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The Mbssbauer Effect 



Frontiers in Physics 

A Lecture Note and Reprint Series 

DAVID PINES, Editor 



N. Bloembergen NUCLEAR MAGNETIC RELAXATION: A Re- 
print Volume 

Geoffrey F. Chew S-MATRIX THEORY OF STRONG INTER- 
ACTIONS: A Lecture Note and Reprint Volume 

R. P. Feynman QUANTUM ELECTRODYNAMICS: A Lecture 
Note and Reprint Volume 

R. P. Feynman THE THEORY OF FUNDAMENTAL PROC- 
ESSES: A Lecture Note Volume 

Hans Frauenfelder THE MOSSBAUER EFFECT: A Collection 
of Reprints with an Introduction 

David Pines THE MANY-BODY PROBLEM: A Lecture Note 
and Reprint Volume 

L. Van Hove, N. M. Hugenholtz, and L. P. Howland PROB- 
LEMS IN THE QUANTUM THEORY OF MANY-PARTICLE 
SYSTEMS 



The 

Mossbauer 

Effect 



A Review— with a Collection of Reprints 



HANS FRAUENFELDER 

University of Illinois 



W. A. BENJAMIN, INC. 

New York 1962 



& PHYSICS 

the mosssba«:r effect 

A Review— with a Collection of Reprints 

Copyright © 1962 by W. A. Benjamin, Inc. 
All rights reserved 

Library of Congress Catalog Card Number: 61-18181 
Manufactured in the United States of America 



W. A. BENJAMIN, INC. 

2465 Broadway, New York 25, New York 



EDITOR'S FOREWORD 



The problem of communicating in a coherent fashion the recent 
developments in the most exciting and active fields of physics 
seems particularly pressing today. The enormous growth in the 
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"point of view" concerning it. Formal monographs cannot meet 
such a need in a rapidly developing field, and, perhaps more im- 
portant, the review article seems to have fallen into disfavor. In- 
deed, it would seem that the people most actively engaged in devel- 
oping a given field are the people least likely to write at length 
about it. 

"Frontiers in Physics" has been conceived in an effort to im- 
prove the situation in several ways. First, to take advantage of the 
fact that the leading physicists today frequently give a series of 
lectures, a graduate seminar, or a graduate course in their special 
fields of interest. Such lectures serve to summarize the present 
status of a rapidly developing field and may well constitute the only 
coherent account available at the time. Often, notes on lectures ex- 
ist (prepared by the lecturer himself, by graduate students, or by 
postdoctoral fellows) and have been distributed in mimeographed 
form on a limited basis. One of the principal purposes of the 
"Frontiers in Physics" series is to make such notes available to 
a wider audience of physicists. 

I 



.. 



vi EDITOR'S FOREWORD 

It should be emphasized that lecture notes are necessarily rough 
and informal, both in style and content, and those in the series will 
prove no exception. This is as it should be. The point of the series 
is to offer new, rapid, more informal, and, it is hoped, more effec- 
tive ways for physicists to teach one another. The point is lost if 
only elegant notes qualify. 

A second way to improve communication in very active fields of 
physics is by the publication of collections of reprints of recent ar- 
ticles. Such collections are themselves useful to people working in 
the field. The value of the reprints would, however, seem much en- 
hanced if the collection would be accompanied by an introduction of 
moderate length, which would serve to tie the collection together 
and, necessarily, constitute a brief survey of the present status of 
the field. Again, it is appropriate that such an introduction be in- 
formal, in keeping with the active character of the field. 

A third possibility for the series might be called an informal 
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step between lecture notes and formal monographs. It would offer 
the author an opportunity to present his views of a field that has 
developed to the point at which a summation might prove extraor- 
dinarily fruitful, but for which a formal monograph might not be 
feasible or desirable. 

Fourth, there are the contemporary classics— papers or lectures 
which constitute a particularly valuable approach to the teaching 
and learning of physics today. Here one thinks of fields that lie at 
the heart of much of present-day research, but whose essentials 
are by now well understood, such as quantum electrodynamics or 
magnetic resonance. In such fields some of the best pedagogical 
material is not readily available, either because it consists of pa- 
pers long out of print or lectures that have never been published. 

"Frontiers in Physics" is designed to be flexible in editorial 
format. Authors are encouraged to use as many of the foregoing 
approaches as seem desirable for the project at hand. The publish- 
ing format for the series is in keeping with its intentions. Photo- 
offset printing is used throughout, and the books are paperbound, in 
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in this country and abroad. 

Finally, because the series represents something of an experi- 
ment on the part of the editor and the publisher, suggestions from 
interested readers as to format, contributors, and contributions 
will be most welcome. 



DAVID PINES 
Urbana, Illinois 
August 1961 



PREFACE 

Only four years ago, Rudolf Mossbauer discovered what is now 
known as the Mossbauer effect: Nuclei that are embedded in solids 
can emit and absorb low- energy gamma rays which display the natu- 
ral line width and possess the full transition energy. No recoil en- 
ergy is transferred to lattice vibrations. Mossbauer' s experiment 
seemed at first to pass unnoticed, but within two years it was re- 
peated and extended. Physicists soon realized that they had at hand 
a new and beautiful tool— simple in its basic ideas, requiring only a 
minimum of equipment, and allowing ingenious applications not only 
in nuclear physics, but also in relativity and in solid-state physics. 
The early trickle of publications became a stream. The rapidly 
growing body of knowledge was discussed and information exchanged 
in two international conferences in 1960 and 1961. In 1961, Rudolf 
Mossbauer received the Nobel prize. 

In the present volume, a number of reprints on the Mossbauer ef- 
fect are collected. I have tried to select those publications which are 
either essential to the understanding of the development of this field 
or which are useful as references for further work. As an introduc- 
tion to these reprints, the first six chapters contain a review of the 
Mossbauer effect. 

During the preparation of the introductory notes I have enjoyed 
many discussions with friends and colleagues. I should like to thank 
particularly K. Bleuler, J. D. Jackson, W. E. Lamb, Jr., R. L. Moss- 
bauer, D. E. Nagle, D. G. Ravenhall, J. R. Schrieffer, and A. H. Taub 
for stimulating remarks. P. Debrunner, D. W. Hafemeister, S. Mar- 
gulies, R. J. Morrison, and D. N. Pipkorn have read through many 
versions of the manuscript and I am grateful to them for forcing me 



vn 



to make it more readable than it might otherwise have been. Finally, 
I should like to thank Mrs. M. Runkel for her unfailing help in the 
preparation of the manuscript. 



HANS FRAUENFELDER 



Urbana, Illinois 
December 1961 



ACKNOWLE DGMENTS 



The publisher wishes to acknowledge the assistance of the follow- 
ing organizations in the preparation of this volume: 

The American Institute of Physics, for permission to reprint the 
articles from the Physical Review, Physical Review Letters, Soviet 
Physics JETP, The Review of Scientific Instruments, and the Journal 
of Applied Physics. 

Academic Press, Inc., for permission to reprint the articles from 
the Annals of Physics. 

Gauthier-Villars Imprimeur- Libra ire, for permission to reprint 
the articles from Comptes rendus hebdomadaires des seances de 
l'academie des sciences. 

The Physical Society, for permission to reprint the articles from 
their Proceedings. 

The Italian Physical Society, for permission to reprint the article 
from II Nuovo cimento. 

Springer Verlag, for permission to reprint the article from 
Zeitschrift fur Physik. 

Verlag der Zeitschrift fur Naturforschung, for permission to 
reprint the article from Zeitschrift fur Naturforschung. 

North-Holland Publishing Co., for permission to reprint the ar- 
ticle from Nuclear Instruments and Methods. 

Macmillan & Co., Ltd., for permission to reprint the article from 
Nature. 



Vlll 



CONTENTS 

Editor's Foreword v 

Preface vii 

1. Introduction 

1-1 Resonance fluorescence 1 

1-2 Cross section for resonance processes 7 

1-3 Mossbauer's discovery 11 

1-4 Literature and reprints 13 

2. Theory 

2-1 Preliminary remarks 14 

2-2 Lattice vibrations 15 

2-3 The classical theory 17 

2-4 The physical picture 20 

2-5 Sketch of the theory 26 

3. Experimental Apparatus and Problems 

3-1 Survey 33 

3-2 Isotopes 36 

3-3 Sources and absorbers 39 

3-4 Apparatus 40 



ix 



x CONTENTS 

3-5 Corrections 44 

3-6 Useful information 47 

4. Nuclear Properties 

4-1 Introduction 50 

4-2 Lifetime and conversion coefficient 51 

4-3 Nuclear moments 51 

4-4 Isomeric shifts 53 

4-5 Parity experiments 57 

5. General Physics 

5-1 Survey 58 

5-2 Gravitational red shift 58 

5-3 Accelerated systems 62 

5-4 Second-order Doppler shift 63 

5-5 Frequency and phase modulation 65 

5-6 The uncertainty relation between energy and time 66 

5-7 Recoilless Rayleigh scattering 69 

5-8 Coherence and interference 71 

5-9 Polarization 73 

6. Solid-State Physics 

6-1 Survey 76 

6-2 Lattice properties 78 

6-3 Internal fields 82 

6-4 Impurities and imperfections 83 

6-5 Low temperatures 84 

7. Bibliography on the Mossbauer Effect 

7-1 Review articles 86 

7-2 Research papers 87 

7-3 Conference reports 96 



CONTENTS xi 

Reprints and Translations 

Section 1 

The Effect of Collisions upon the Doppler Width of Spectral 

Lines, by R. H. Dicke • Phys. Rev., 89, 472 (1953) 99 

Kernresonanzfluoreszenz von Gammastrahlung in Ir 191 , 

by R. L. Mossbauer • Z. Physik, 151, 124 (1958) 101 

Theoretical part of the preceding paper in English translation 121 

Kernresonanzabsorption von y-Strahlung in Ir 191 , by R. L. 
Mossbauer • Naturwissenschaften, 45, 538 (1958) 
(English translation) 127 

Kernresonanzabsorption von y-Strahlung in Ir 191 , by R. L. 

Mossbauer • Z. Naturforsch., 14a, 211 (1959) 130 

Capture of Neutrons by Atoms in a Crystal, by W. E. Lamb, Jr. 

• Phys. Rev., 55, 190 (1939) 136 

Section 2 

Study of Lattice Vibrations by Resonance Absorption of 
Nuclear Gamma Rays, by W. M. Visscher • 
Ann. Phys., 9, 194 (1960) 144 

Some Simple Features of the Mossbauer Effect, by 

H. J. Lipkin • Ann. Phys., 9, 332 (1960) 161 

Resonance Radiation of Nuclei Bound in a Lattice, 

by D. R. Inglis (original paper) 169 

Resonance Absorption of Nuclear Gamma Rays and the 
Dynamics of Atomic Motions, by K. S. Singwi and 
A. Sjolander • Phys. Rev., 120, 1093 (1960) 194 

Diffusion des photons sur les atomes et les noyaux dans 
les cristaux, by C. Tzara • J. phys. radium, 22, 
303 (1961) 204 

Section 3 

Diffusion resonante du rayonnement y de 23.8 kev de Sn 119 
6mis sans recul, by R. Barloutaud, J. L. Picou, and 
C. Tzara • Compt. rend., 250, 2705 (1960) 2Q9 

Zeeman Effect in the Recoilless y-Ray Resonance of Zn 67 , 
by P. P. Craig, D. E. Nagle, and D.R. F. Cochran • 
Phys. Rev. Letters, 4, 561 (1960) 212 

Mossbauer Effect in Ferrocyanide, by S. L. Ruby, L. M. 

Epstein, andK.H.Sun • Rev.Sci. Instr., 31,580(1960) 216 



xii CONTENTS 



Transmission and Line Broadening of Resonance Radiation 
Incident on a Resonance Absorber, by S. Margulies 
and J. R. Ehrman • Nuclear Instr. and Methods, 12, 
131 (1961) 218 



Section 4 

Polarized Spectra and Hyperfine Structure in Fe 57 , by 

5. S. Hanna, J. Heberle, C. Littlejohn, G. J. Perlow, 
R. S. Preston, and D. H. Vincent • Phys. Rev. 

Letters, 4, 177 (1960) 225 

Evidence for Quadrupole Interaction of Fe 57m and Influence 
of Chemical Binding on Nuclear Gamma-Ray Energy, 
by O. C. Kistner and A. W. Sunyar • Phys. Rev. 
Letters, 4, 412 (1960) 229 

Electric Quadrupole Splitting and the Nuclear Volume 
Effect in the Ions of Fe 57 , by S. DeBenedetti, 
G. Lang, and R. Ingalls • Phys. Rev. Letters, 

6, 60 (1961) 233 

Interpretation of the Fe 57 Isomer Shift, by L. R. Walker, 
G. K. Wertheim, and V. Jaccarino • Phys. Rev. 
Letters, 6, 98 (1961) 236 

Section 5 

Apparent Weight of Photons, by R. V. Pound and G. A. 

Rebka, Jr. • Phys. Rev. Letters, 4, 337 (1960) 240 

Measurement of the Red Shift in an Accelerated System 
Using the Mossbauer Effect in Fe 57 , by H. J. Hay, 
J. P. Schiffer, T. E. Cranshaw, and P. A.Egelstaff 
• Phys. Rev. Letters, 4, 165 (1960) 245 

Absence of Doppler Shift for Gamma Ray Source and 

Detector on Same Circular Orbit, by D. C. Champeney 

and P. B. Moon • Proc. Phys. Soc. (London), 77, 350 

(1961) 247 

Variation with Temperature of the Energy of Recoil-free 
Gamma Rays from Solids, by R. V. Pound and G. A. 
Rebka, Jr. • Phys. Rev. Letters, 4, 274 (1960) 250 

Temperature -Dependent Shift of y Rays Emitted by a 

Solid, by B. D. Josephson • Phys. Rev. Letters, 4, 

341 (1960) 252 

Acoustically Modulated y Rays from Fe 57 , by S. L. Ruby and 

D.I. Bolef • Phys. Rev. Letters, 5, 5 (1960) 253 



CONTENTS xiii 

Measurement of the Refractive Index of Lucite by 

Recoilless Resonance Absorption, by L. Grodzins 

and E. A. Phillips • Phys. Rev., 124, 774 (1961) 256 

Time Dependence of Resonantly Filtered Gamma Rays 
from Fe 57 , by F. J. Lynch, R. E. Holland, and 
M. Hamermesh • Phys. Rev., 120, 513 (1960) 259 

Effect of Radiofrequency Resonance on the Natural Line 
Form, by M. N. Hack and M. Hamermesh • Nuovo 
cimento, 19, 546 (1961) 267 

Recoilless Rayleigh Scattering in Solids, by C. Tzara and 

R. Barloutaud • Phys. Rev. Letters, 4, 405 (1960) 279 

Sur la possibilite de mettre en evidence la coherence de 
phase dans la diffusion de resonance des rayons y 
par des noyaux atomiques, by A. Kastler • Compt. 
rend., 250, 509 (1960) 281 

Resonant Scattering of the 14-kev Iron- 57 y-Ray and Its 
Interference with Rayleigh Scattering, by P. J. 
Black and P. B. Moon • Nature, 188, 481 (1960) 284 

Section 6 

The Mossbauer Effect in Tin from 120°K to the Melting 
Point, by A.J. F. Boyle, D. St. P. Bunbury, 
C„ Edwards, and H. E. Hall • Proc. Phys. Soc. 
(London), 77, 129 (1961) 288 

Mossbauer Effect: Applications to Magnetism, by G.K. 

Wertheim« J. Appl.Phys., Suppl., 32, No. 3, 110S (1961) 295 

Direction of the Effective Magnetic Field at the Nucleus 
in Ferromagnetic Iron, by S. S. Hanna, J. Heberle, 
G. J. Perlow, R. S. Preston, and D. H. Vincent • 
Phys. Rev. Letters, 4, 513 (1960) 303 

Temperature Dependence of the Internal Field in Ferro- 
magnets, by D. E. Nagle, H. Frauenfelder, R. D. 
Taylor, D. R. F. Cochran, and B. T. Matthias • 
Phys. Rev. Letters, 5, 364 (1960) 306 

Polarization of the Conduction Electrons in the Ferro- 
magnetic Metals, by A. J. F. Boyle, D.St. P. Bunbury, 
and C. Edwards • Phys. Rev. Letters, 5, 553 (1960) 308 

Hyperfine Field and Atomic Moment of Iron in Ferromag- 
netic Alloys, by C.E.Johnson, M.S.Ridout, T.E. 
Cranshaw, and P.E. Madsen • Phys. Rev. Letters, 
6, 450 (1961) 312 



xiv CONTENTS 

Internal Magnetic Fields in Manganese -Tin Alloys, by 
L. Meyer -Schtitzmeister, R. S. Preston, and 
S. S. Hanna • Phys. Rev., 122, 1717 (1961) 314 

Study of the Internal Fields Acting on Iron Nuclei in Iron 
Garnets, Using the Recoil-free Absorption in Fe 57 
of the 14.4-kev Gamma Radiation from Fe 57m , by 
R. Bauminger, S. G. Cohen, A. Marinov, and 
S. Ofer • Phys. Rev., 122, 743 (1961) 318 

On the Use of the Mossbauer Effect for Studying Localized 
Oscillations of Atoms in Solids, by S. V. Maleev • 
JETP, 39, 891 (1960) (in Russian); Soviet Physics 
JETP, 12, 617 (1961) (in English). English transla- 
tion reprinted here 324 

Polarization of Co 57 in Fe Metal, by J. G. Dash, 
R. D. Taylor, D. E. Nagle, P. P. Craig, and 
W. M. Visscher • Phys. Rev., 122, 1116 (1961) 325 

Comments and Corrections 335 



THE MOSSBAUER 
EFFECT 



L 




INTRODUCTION 

1-1 RESONANCE FLUORESCENCE 

The beauty and fascination of physics rarely becomes more ap- 
parent than when one follows a particular topic through the various 
stages of its development. The history of the neutrino, the most 
elusive of all particles, is an example. Another is the story of res- 
onance fluorescence, the subject of the present volume. In this chap- 
ter, we outline briefly some phases in the history of atomic and nu- 
clear resonance fluorescence and the Mossbauer effect, which begin 
with the mechanical interpretation of matter and end, hopefully only 
temporarily, with the Mossbauer effect. 

The story of resonance fluorescence starts at the end of the last 
century with Lord Rayleigh, who suggested that resonance scattering 
should occur in atomic systems. Considerable time elapsed after 
this prediction before R. W. Wood discovered resonance radiation in 
1904. The explanation of resonance scattering was then based en- 
tirely on mechanical analogies. The scattering resonators were as- 
sumed to be exactly in tune with the frequency of the incoming radi- 
ation. Wood's discovery led to other experiments, the techniques 
were rapidly improved, and a wealth of data was accumulated. 1 4 1 



t References not directly connected with the Mossbauer effect are 
designated by superscripts and listed at the bottom of each page. 
References dealing specifically with the Mossbauer effect will be 
found listed in Chapter 7; they are indicated in the text by the name 
of the first author and the year of publication. Papers reprinted 
are denoted by "Reprint" and the name of the first author. 

X R. W. Wood, "Physical Optics," Macmillan, New York, 1934, 
3rd ed., Chap. XVIII. 

2 A. C. G. Mitchell and M. W. Zemansky, "Resonance Radiation 



2 THE MOSSBAUER EFFECT 

It is interesting to note that, in the mechanical picture underlying 
these early experiments, incoming and outgoing radiation maintain 
a fixed phase relation. 

Resonance fluorescence fitted well into the Bohr theory, which 
superseded the earlier pictures: An atom, decaying from an excited 
state B to its ground state A, emits a photon of well-defined fre- 
quency <j) Y . When such a photon passes through a gas consisting of 
the same element as the emitter, it can be absorbed and excite a 
target atom into the state B. After a short time, this excited target 
atom will in turn decay and emit a photon of frequency a> r . Primary 
and secondary radiation thus have the same frequency, but the proc- 
esses of absorption and reemission are independent and no fixed 
phase relation exists between them. 

Many of the aspects of resonance radiation were correctly de- 
scribed by the Bohr theory and by the early quantum mechanics. 
However, problems connected with line width, radiation damping, 
and coherence were not easily explained. The Dirac theory of radi- 
ation, in the hands of Weisskopf and Wigner, finally provided a com- 
plete description of the processes of emission, absorption, and res- 
onance fluorescence. 5 ' 8 All the fundamental aspects of resonance 
radiation seemed to be solved, interest in basic investigations of 
atomic resonance fluorescence decreased, and research moved on to 
more complicated problems of fluorescence and phosphorescence. 3 ' 4 
(incidentally it may be noted that atomic resonance has recently 
played an important role once more in the optical orientation of nu- 
clei.) 

Since atomic resonance radiation depends essentially on the ex- 
istence of quantized levels and since quantized levels also occur in 
nuclei, the possibility of observing nuclear resonance fluorescence 
was obvious, and the search was started in 1929 by Kuhn. 7 However, 
even though problems in atomic and nuclear resonance seem very 
similar, there exist marked differences which render nuclear ex- 
periments much more difficult. In order to outline these difficulties, 
the process of resonance fluorescence must be discussed in more 

and Excited Atoms," Cambridge University Press, Cambridge, 1934. 

3 P. Pringsheim, "Fluorescence and Phosphorescence,' ' Inter- 
science, New York, 1949. 

4 J. G. Winans and E.J. Seldin, Fluorescence and Phosphores- 
cence, in E. U. Condon and H. Odishaw (eds.), "Handbook of Physics,' ' 
McGraw-Hill, New York, 1958. 

5 V. Weisskopf, Ann. Physik, 9, 23 (1931). V. Weisskopf and 
E. Wigner, Z. Physik, 63, 54 (1930); 65, 18 (1930). 

8 W. Heitler, "Quantum Theory of Radiation," Clarendon Press, 
Oxford, 1949. 

7 W. Kuhn, Phil. Mag., 8, 625 (1929). 



INTRODUCTION 3 

detail. Consider a free atomic or nuclear system, of mass M, with two 
levels A and B, separated by an energy E r - If the system decays from 
B to A by emission of a photon of energy E , momentum conservation 
demands that the momentum p of the photon and the momentum P of 
the recoiling system be equal and opposite. The recoiling system 
hence receives an energy R, given by 



P 2 p 2 Ey 
2M 2M 2Mc 2 



(1) 



In this derivation, it is assumed that the recoiling system can be 
treated nonrelativistically— an assumption that is extremely well 
satisfied in atomic and nuclear spectroscopy. Actually, all gamma 
rays involved in atomic and nuclear resonance fluorescence have en- 
ergies that are small compared to the rest energy Mc 2 of the emit- 
ting atomic or nuclear system. The recoil energy is thus very small 
compared to the gamma- ray energy. 

Energy conservation connects E r , E y , and R: 

E r = E y + R (2) 

Since R is very small compared to Ey, and since, as will be obvious 
later, one needs to know R only to moderate accuracy, Ey can be re- 
placed by the transition energy E r in (l): 

R = E 2 /2Mc 2 (10 

The recoil energy R is plotted in Fig. 1-1 as a function of the transi- 
tion energy E r for a quantum system of mass number 100. 

One more feature now enters the discussion, namely, the width of 
the excited state B. Assume that the mean life against decay of the 
state B is r. According to the Heisenberg uncertainty relation, the 
energy in state B then cannot be measured sharply, but only within 
an uncertainty given by 

r-r=R (3) 

where 27rn" is Planck's constant. A detailed examination employing 
perturbation theory shows indeed that the decaying state B cannot 
be characterized by one well-defined energy E r only, but that the en- 
ergy E of the state is distributed about the center energy E r as shown 
in Fig. l-2a. 6 The energy of the stable ground state, according to (3), 
is sharp. Photons emitted in the transition from B to A thus show a 
distribution in energy Ey, centered around E r - R, and displaying a 
" natural line shape" of width T (Fig. l-2b). To give an order 
of magnitude, the line width T corresponding to a mean life r = 10~ 8 



THE MOSSBAUER EFFECT 




10" 1 



1 ev 



Y 



optical transitions 



1 kev 1 Mev 

f- — atomic X rays — -| 

[• nuclear gamma rays — *- 



-E, 



Fig. 1-1 Recoil energy R and Doppler broadening D 
(for a gas at 300 °K) as a function of the 
transition energy E r for a nucleus of mass 
number 100. For comparison, the natural 
line width T corresponding to a lifetime of 
10 " 8 sec is shown also. 



sec is shown in Fig. 1-1. Such a mean life is typical for atomic 
states and net -unreasonable for low- lying nuclear levels. 

Returning to resonance fluorescence again, when a photon of en- 
ergy Ey and momentum p strikes a target of mass M, which is ini- 
tially at rest, the entire momentum p is transferred to the target. 
The target thus recoils and the energy of recoil R is again given by 
(l) or (!'). This energy must be supplied by the gamma ray. Thus 



INTRODUCTION 




E r - R 





Fig. 1-2 Energy distributions involved in resonance 
fluorescence, (a) Energy distribution of ex- 
cited state B. (b) Energy distribution of 
photons emitted in a transition B — A. 
(c) Energy spectrum required to excite 
state B in target and provide center-of- 
mass energy R. (d) Overlap of M and (T. 



only an energy E r - R is available for the excitation of internal de- 
grees of freedom (Fig. l-2b). In order to excite a level of energy E r , 
the incoming gamma ray must have an energy E r + R, as shown in 
Fig. l-2c. Resonance fluorescence can occur only if some of the 
incoming photons possess enough energy to "reach" the state B and 
at the same time provide the energy R to recoiling system. Thus, 



6 THE MOSSBAUER EFFECT 

only the overlapping part of the spectra 2b and 2c is responsible for 
resonance fluorescence (Fig. l-2d). The condition for overlap is 

2R < T (4) 

A glance at Fig. 1-1 shows that optical transitions fulfill condition (4), 
but that nuclear transitions are far from it. 

In the discussion so far, the emitting and the absorbing system 
were assumed to be at rest. Actually, however, source and target 
atoms are in thermal motion, and this motion introduces an additional 
widening of the emission and absorption lines, called Doppler broad- 
ening. In order to see this effect qualitatively, consider a source 
with initial momentum P i? which emits a photon of momentum p. The 
momentum of the source after the emission is given by P^ - p. The 
energy gained by the source and hence lost by the gamma ray is, in 
good approximation, t 

2M 2M 2M M vo; 

The first term on the right-hand side of (5) is the recoil energy R 
[Eq. (l)] of the initially stationary system; the second term repre- 
sents the Doppler broadening. Introducing the kinetic energy 
e = P|/2M of the quantum system before the photon emission and 
a "Doppler energy" 

D = 2(8R) 1/2 (6) 

this second term can be written 

D cos <p 

where <p denotes the angle between p and P^ The photon energy then 
becomes 

E = E r - R' = E r - R + D cos (p (7) 

where E r = Eg - E» is the total energy released in the transition 

B — A. Generally the angle <p will vary from to 277 and the Doppler 
term in (7) causes a spread in the photon energy E^ of the order of D. 

tin (5) it is assumed that the mass of the quantum system does 
not change during the emission. Actually, there is a very small 
change in mass, because the photon carries away energy. This effect 
has been observed experimentally (Sec. 5-4) but is unimportant in 
the present context. 



INTRODUCTION 7 

In a source in which the velocities of the individual emitters are 
isotropic in direction and Maxwellian in magnitude, this additional 
broadening is of the order 

D a 2(cR) 1/2 (8) 

where e is the average value of the kinetic energy of the emitters. In 
Fig. 1-1 curve D represents such a Doppler broadening, correspond- 
ing roughly to that of a gaseous source at room temperature. Com- 
parison of Fig. 1-2 and the three curves in Fig. 1-1 shows that the 
natural line width T is not always the dominant feature in resonance 
fluorescence; the Doppler broadening also plays an important role. 
For optical radiation, the recoil energy R is small compared to the 
Doppler broadening, emission and absorption lines overlap, and res- 
onance conditions are obtained. For nuclear gamma rays, however, 
the recoil energy is comparable to, or greater than, the Doppler 
broadening, and the discussion of when to expect a measurable res- 
onance fluorescence effect requires an investigation of the cross 
section for resonance fluorescence under various conditions. 

1-2 CROSS SECTION FOR RESONANCE PROCESSES 

Resonance experiments with gamma rays are usually performed 
by either measuring the scattered intensity (resonance fluorescence 
or resonance scattering) or by determining the attenuation of a beam 
due to resonance absorption. The cross sections for these two proc- 
esses, for an incident gamma ray of energy E and wavelength 27T* 2 
can be calculated in a straightforward way 8 ' 8 and, for thin absorbers, 
can be written 

r 2 

(9) 

(10) 



In these expressions, T is the total width of the absorption line, T y 
its gamma- ray width, and cr the maximum resonance cross section 
given by 



*scatt (E) 
"abs (E) = 




r 2 




" a ° 4(E 


- E r ) 2 

riy 


+ r 2 


a o 4(E - 


E r ) 2 + 


r 2 



2I B + 1 
2I A +1 



27T* 2 (11) 



In (11) I A is the spin of the ground state A and Ig the spin of the ex- 
cited state B. 



J. D. Jackson, Can. J. Phys., 33, 575 (1955). 



8 THE MOSSBAUER EFFECT 

Scattering and absorption cross sections as given by (9) and (10) 
show a characteristic energy dependence of the form 

I(E) = 2i (E-E r ) 2 + [(l/2)rp (12) 

which is normalized to 

oo 

/ 1(E) dE = 1 (12') 

o 

Corresponding curves are sketched in Fig. 1-2. These distributions 
are said to show a Breit-Wigner or Lorentz shape. The parameter 
T gives the full width of the distribution at half maximum. 

Since expressions (9) and (10) are fundamental for the Mossbauer 
effect, it is important to know that they are derived under the fol- 
lowing assumptions: 

a. Only one absorbing or scattering level exists. If more than 
one state appears at, or close to, the energy E , or if the state at E r 
is split into sublevels, for instance by a strong magnetic field, then 
(9) and (10) must be modified (Margulies, 1962). 

b. The width of the level is given entirely by decay processes. In 
nuclei, in all interesting cases, the two competing modes are gamma- 
ray emission and internal conversion. Total width and gamma-ray 
width are then related by the equation 

r r = lh r (13) 

where a is the coefficient of internal conversion. If the level is 
broadened by other influences, the line shape need not be Lorentz ian. 

c. The incoming photon is monoenergetic. An actual source, how- 
ever, will possess an energy spectrum l(E), where l(E) dE denotes 
the number of gamma rays emitted with energies between E and 

E + dE. The observed cross sections for scattering and absorption 
then become integrals of the form 

oo 



/a(E)l(E) dE 



/ X 

55 (14) 

J 1(E) dE 
o 

Here in the introduction only two special cases of (14) will be treated. 
Further cases appear in the various reprints. 

1. The incoming gamma ray possesses an energy E r ; its width is 
small compared to the width T, and the conversion coefficient a is 
zero. I(E) in this case is essentially a delta function 6(E r ), and from 
(9), (10), (13), and (14) one finds 



INTRODUCTION 



°scatt r abs r ° 

This result explains the designation "maximum resonance cross 
section" for a . 

2. The incoming gamma ray has an energy spectrum l(E) de- 
scribed by a Lorentz shape [Eq. (12)], with a width r identical to the 
width of the absorbing state, and with its energy centered at E r . In 
a resonance fluorescence experiment these assumptions mean that 
the recoil energy R is either negligible or has been compensated 
for, and that emitting and absorbing state are identical and are not 
broadened by external influences. With these assumptions, one finds 
from (10), (12), and (14) for the maximum cross section for the res- 
onance absorption of a gamma ray with width r by a level with iden- 
tical width, 

r a 

a eff abs = a ° 2T = 2(1+ a) (l5) 

One trivial remark must be added here. The actual width of the in- 
coming gamma ray and of the absorbing level are both given by r and 
not by T y . TV is not a width that can be traced experimentally, but 
rather a quantity F which characterizes the fraction of decays pro- 
ceeding by photon emission. 

Equation (15) leads back to resonance fluorescence. In fact, it 
describes resonance absorption under the most ideal conditions, 
namely, (i) emission and absorption line possess the natural line 
width, and (il) emission and absorption line are centered at the same 
energy. If these two conditions are satisfied, the cross section for 
resonance absorption, given by (15), is very large provided the con- 
version coefficient is reasonably small. The effective maximum 
resonance cross section is plotted against photon energy in Fig. 1-3. 
It is obvious from the values in Fig. 1-3 that resonance experiments 
should yield large effects provided the two conditions I and n can be 
satisfied. Another glance at Fig. 1-1 shows that for optical reso- 
nance fluorescence, the second assumption is valid, but the first is 
not because Doppler broadening considerably widens the lines. The 
correct cross section for resonance scattering in the presence of 
Doppler broadening can be calculated easily. Even without calcula- 
tion, one can make a good guess. Assume the target to be at rest 
and the incoming photons to have an energy spectrum characterized 
by a Doppler width D. Only a small part of the widened incoming 
spectrum overlaps with the resonance level. The fraction of incom- 
ing photons capable of undergoing resonance absorption is approxi- 
mately equal to r/D. This fraction, however, is centered at the res- 
onance energy E r and hence enjoys approximately the maximum res- 
onance cross section (J Ty/r given by (10) for E = E r . The effective 
absorption cross section thus becomes 



10 



THE MOSSBAUER EFFECT 



d r r r r > 



(16) 



This estimate, which is borne out quite well by the exact calcula- 
tion, shows that the maximum cross section is reduced by a factor 



7i* 2 , cm 2 




ev 



Fig. 1-3 Effective maximum resonance absorption 

cross section n-k 2 as a function of photon en- 
ergy. The curve is valid for vanishing con- 
version coefficient and for identical spins in 
the two states involved. To find the cross 
section for a decay with spins 1^, Ig, and 
with conversion coefficient a, multiply the 
value obtained from the curve by 

2I B + 1 1 



2I A + 1 1 + a 



INTRODUCTION 11 



IV /D. For optical transitions, internal conversation is impossible, 
and T y = T. From Fig. 1-1 one can see that the maximum cross sec- 
tion is reduced only by one or two orders of magnitude and that ex- 
periments with optical transitions should be feasible— as indeed they 
are. 

In nuclear experiments, both conditions are violated, but the sec- 
ond can be restored by several means. In 1950, Moon performed the 
first successful experiment by plating a source of Au 198 onto the tips 
of a steel rotor and spinning the rotor up to tip speeds of 800 m/sec. 9 
Gamma rays emitted tangentially thus gained additional energy suf- 
ficient to compensate for the recoil loss. Other possibilities of com- 
pensating for the recoil energy loss, for instance the use of the recoil 
from a transition preceding the gamma ray to be investigated, or the 
heating of source and absorber to increase the average kinetic en- 
ergy, have also been used with positive results. 

Despite these ingenious experiments, classical nuclear resonance 
fluorescence experiments are never easy, since condition (I) is not 
fulfilled. The ratio Ty/D in (18) is much smaller for nuclear than 
for optical transitions. Figure 1-1 indicates that the maximum cross 
section is decreased by many orders of magnitude. Despite these 
difficulties many good experiments have been performed since 1950. 
These experiments and the underlying theory are discussed in a num- 
ber of review articles. 10 " 12 



1-3 MOSSBAUER'S DISCOVERY 

Comparing (15) and (16) one is struck by the price one must pay 
for the effects of thermal motion. The question arises of whether 
there is a way to avoid the large decrease in cross section which is 
caused entirely by nonnuclear processes. 

Dicke in 1952 theoretically studied the reduction of Doppler 
broadening by collisions in a dense gas 13 (Reprint Dicke). As a 
simple model he used a radiating atom enclosed in a one-dimen- 
sional well oscillating back and forth between the walls. Actually, 
such a model is more appropriate to the description of a solid. In 
any case, he found a normal Doppler distribution plus a sharp, non- 
Doppler broadened and unshifted line. Today, such a line would be 



9 P. B. Moon, Proc. Phys. Soc, 63, 1189 (1950). 

10 K. G. Malmfors in K. Siegbahn (ed.), "Beta- and Gamma-Ray 
Spectroscopy/' North- Holland, Amsterdam, 1955, Chap. XVIIl(ll). 

11 F. R. Metzger in O. R. Frisch (ed.), "Progress in Nuclear 
Physics," Pergamon, New York, 1959, Vol. 7. 

12 S. Devons in Fay Ajzenberg-Selove (ed.), "Nuclear Spectro- 
scopy," Part A, Academic, New York, 1960, Chap. IV(B). 
13 R. H. Dicke, Phys. Rev., 89, 472 (1953). 



12 THE MOSSBAUER EFFECT 

called a Mossbauer line. No successful experiments were carried 
out, however, before 1957 and Dicke's calculation had no influence on 
the later development. 

R. L. Mossbauer, in 1957, started the dramatic next step from a 
completely different side, by an accidental discovery. (Reprint Moss- 
bauer 1958, 1959) He investigated the nuclear resonance scattering 
of the 129-kev gamma ray from Ir 191 . For this transition, the free 
recoil energy R is 0.05 ev, the Doppler broadening at room temper- 
ature about 0.1 ev. At room temperature emission and absorption 
spectrum thus overlap considerably, and resonance scattering can be 
observed. In order to reduce this residual scattering, Mossbauer 
cooled both source and absorber and expected a decrease in effect. 
Instead, the resonance scattering increased! After carefully ascer- 
taining that this increase was not spurious, Mossbauer set out to 
investigate the nature of this surprising effect. Following a sug- 
gestion by Jensen, he found a paper by Lamb 14 (Reprint Lamb) in 
which the effect of lattice binding on the capture cross section of 
slow neutrons is discussed. The figures 2 and 3 in Lamb's paper 
already show peaks that could be called "Lamb- Mossbauer peaks." 
(Compare also reference 15.) By adapting Lamb's calculation to 
gamma rays, Mossbauer was able to explain his own experimental 
results. In one beautiful experiment, he had thus found the solution 
to both problems outlined in Sec. 1-2: A fraction of the 129-kev 
gamma rays, emitted by the cooled source, did not show a measur- 
able recoil energy loss, and these same gamma rays also did not 
display a Doppler broadening. Their line width corresponded to the 
natural line width. 

After Mossbauer' s first publication (Reprint Mossbauer 1958), 
about a year elapsed before other laboratories, particularly Los 
Alamos and Argonne, began to repeat and extend his experiments. 
(As an amusing side remark, it may be told that the research at Los 
Alamos was started by a bet. Two physicists were discussing Moss- 
bauer' s discovery and one of the two did not believe it. The other 
bet a nickel that the effect was real and that he could repeat the ex- 
periment. He won.) All these early experiments, performed with 
Ir 191 , were complicated by low- temperature requirements and by the 
smallness of the effect. 

The next major advance came through the discovery of the Moss- 
bauer effect in Fe 57 , made independently at Harvard, Harwell, the 
University of Illinois, and Argonne. The ease with which the effect 
can be demonstrated with Fe 57 , its very large size, its persistence 
up to temperatures of over 1000 °C, and the very narrow natural line 
width, immediately changed this field of physics from one accessible 



14 W. E. Lamb, Jr., Phys. Rev., 55, 190 (1939) 
15 H. Ott, Ann. Physik, 23, 169 (1935). 



INTRODUCTION 13 

to only a few laboratories to one in which even modestly equipped 
groups could compete. Moreover, the extremely narrow line width 
of Fe 57 and the fantastically large value of E r /r opened possibilities, 
such as the measurement of the red shift of photons in a terrestrial 
laboratory, which were thought to be impossible a short time earlier. 

After these advances, more nuclei showing recoilless resonance 
absorption were found and many new and exciting experiments were 
performed. The various phases in the still young history of the 
Mossbauer effect are summarized in Table 1-1, due to H. J. Lipkin. 

Table 1-1 History of the Mossbauer Effect 3, 



Period 


Date 


Remarks 


Prehistoric 


Before 1958 


Might have been discovered, 
but wasn't 


Early iridium age 


1958 


Discovered, but not noticed 


Middle iridium age 


1958-1959 


Noticed, but not believed 


Late iridium age 


1959 


Believed, but not interesting 


Iron age 


1959-1960 


Wow!! 



H. J. Lipkin, private communication. 



1-4 LITERATURE AND REPRINTS 

The description of resonance fluorescence in this chapter should 
give some of the background which is usually hard to find in re- 
search papers. Chapters 2 to 6 are more condensed, since they 
are intended mainly as guides to the collected reprints and to the 
literature. Only aspects that seem most important to the understand- 
ing of the physics of Mossbauer effect are given. Further details and 
summaries can be found in the review papers on recoilless emission 
of gamma rays, listed in Chapter 7. 

The reprints collected in the back have been selected either be- 
cause they have been essential in the development of the field or be- 
cause they illustrate important aspects and are useful for contem- 
plating new experiments or extensions of the theory. Any such 
selection is arbitrary and subject to Dyson's law: "If a book is to 
be published at a time T, and if it is supposed to be up-to-date to a 
time T - t, then it will inevitably be out-of-date at the time T + t." 16 
However, it is hoped that within the limits of this law the selection 
and the introductory chapters will prove to be useful. 



16 



F. J. Dyson, Physics Today, 8 (6), 27 (1955) 




THEORY 

2-1 PRELIMINARY REMARKS 

Mossbauer, after his experimental discovery of recoilless gamma- 
ray emission and absorption, successfully explained these phenomena. 
He did so by adapting to gamma rays Lamb's theory of neutron cap- 
ture by atoms bound in a crystal (Reprint Lamb, Reprint and Transla- 
tion Mossbauer). Since then a number of theoretical papers have ex- 
tended and elaborated the basic aspects. Despite the number of pub- 
lications, however, some unsolved and unclear questions remain, as 
illustrated for instance by two quotes from recent papers: "It turns 
out that the existence of the Mossbauer line is a purely classical ef- 
fect,'' and "The Mossbauer effect can only be understood quantum 
mechanically. The classical picture fails completely if one wants to 
understand the behavior of the emitter." Obviously, these two state- 
ments are not in complete agreement. However, they can be recon- 
ciled by showing that the classical treatment indeed yields some of 
the essential features, particularly the existence of an unshifted line, 
but that the complete treatment of the Mossbauer effect, including re- 
coil phenomena, requires quantum mechanics. 

Indeed, a complete treatment of the Mossbauer effect requires 
nearly all of quantum mechanics and it is then not surprising to ob- 
serve that the apparent paradoxes of quantum theory, such as the 
famous double-slit experiment, appear again, in a somewhat changed 
form. In discussing the formulas describing recoilless emission and 
absorption, one must watch out not to quarrel about aspects that can- 
not be measured in principle. No better warning can be given than 
Pauli's: 17 "It is essential to observe that a statement, according to 



17 W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik, 
"Encylopedia of Physics," Springer, Berlin, 1958, Vol. V, Part 1, 
p. 69 (note also page 136). 

14 



THEORY 15 

which a system, independent of a determination by a measurement, 
contains a well-defined internal energy E n , or equivalently is in a 
well-defined stationary state, easily leads to contradictions. This is 
true particularly where the older quantum theory talks of 'transition 
processes' among the various stationary states of the system." 

In this chapter a short outline of the main features of lattice vibra- 
tions (Sec. 2-2) is followed by a discussion of the classical theory 
(Sec. 2-3). A primitive picture underlying the quantum mechanical 
treatment is sketched in Sec. 2-4. Some aspects of the theory usually 
assumed to be known are presented in Sec. 2-5. Finally, some re- 
marks about the De bye- Waller factor are added. 

In all sections, the emphasis is placed on outlining the physical 
features and providing some elementary discussions which are omit- 
ted in the published papers. The complete treatment can be found in 
the reprints on theory. 



2-2 LATTICE VIBRATIONS 

Some knowledge of the theory of lattice vibrations is essential for 
an understanding of the mechanism of recoilless gamma-ray emis- 
sion. The very simplest aspects are presented here as a reminder. 
Detailed treatments are contained, for instance, in Refs. 18-21. 

One of the puzzles of classical physics was the decrease of the 
specific heat of solids below a certain critical temperature. Einstein, 
in 1907, first explained this decrease by assuming that a solid con- 
sisted of a large number of independent linear oscillators, each vi- 
brating with a frequency u^. Tne corresponding spectrum of lattice 
vibrations is shown in Fig. 2- la. 

Einstein's theory explained the decrease of specific heats qualita- 
tively, but the exponential behavior at very low temperatures pre- 
dicted by his theory differed from the experimentally observed T 3 de- 
pendence. Debye, in 1912, derived the T 3 dependence and thus im- 
proved the agreement between theory and experiment by introducing 
a continuum of oscillator frequencies, ranging from zero to a maxi- 
mum frequency oo^ and obeying a distribution function. 

c(u>) = const, u> 2 (17) 



18 M. Blackman in "Encyclopedia of Physics," Springer, Berlin 
1955, Vol. VII, Part 1. 

19 J. DeLauney in F. Seitz and D. Turnbull (eds.), "Solid State 
Physics," Academic, New York, 1956, Vol. 2, p. 219. 

20 C. Kittel, "Introduction to Solid State Physics," Wiley, New 
York, 1956, Chaps. 5 and 6. 

21 E. W. Montroll in E. U. Condon and H. Odishaw (eds.), "Hand 
book of Physics," McGraw-Hill, New York, 1958, Sec. 5-150. 



16 



THE MOSSBAUER EFFECT 



c(w) 



»c(w) 



(a) 



(J0^ 




c(w) 



w n oo 




5x lO^sec- 1 ^ 



Fig. 2-1 Spectrum of lattice vibrations in a solid: (a) Einstein 
model, (b) Debye model, (c) Born- von Karman model. 
[After G. Leibfried and W. Brenig, Z. Physik, 134, 451 
(1953).] 

Such a distribution, shown in Fig. 2- lb, can be derived by assuming 
the solid to be a homogeneous and isotropic medium, the group ve- 
locity of waves of all frequencies to be the same, and the total num- 
ber of (one-dimensional) linear oscillators to be equal to three times 
the number N of atoms in a solid. 

Debye's theory explained the experimental data very well. How- 
ever it was soon realized that the actual vibrational spectra, while 
obeying a go 2 dependence at low frequencies, markedly deviate from 
it at higher frequencies and calculations to find more detailed spec- 
tra were performed. The way to perform these calculations had al- 
ready been pointed out in 1913 by Born and von Karman. They deter- 
mined the spectrum by first finding the normal modes of a lattice 
consisting of mass points connected by springs and by deriving the 
spring constants from the interatomic force law. A result of a typi- 
cal calculation is shown in Fig. 2-lc. The deviation from the Debye 
model is apparent. 

One often assumes for simplicity that the Debye model is correct 
and then defines a characteristic temperature, called Debye temper- 
ature Or> by the equation 



E D =1iu> D = ke D 



(18) 



Here cod is the cut-off frequency shown in Fig. 2- lb, which in the 
Debye theory is given by the condition that the total number of all os- 
cillators be equal to 3N. Experimentally, one can determine the Debye 
temperature from specific heat measurements, from X-ray reflection, 
or from elastic constants. Values range from 88°Kfor Pb to 1000°K 
for Be. However, one must be careful not to identify, without detailed 
investigations, the Debye temperature measured in a conventional way 
with the constant that enters calculations of the Mossbauer effect. 



THEORY 17 

2-3 THE CLASSICAL THEORY 

A classical expression for the intensity of the Mossbauer line has 
been given by Shapiro (Shapiro 1961) and Van Kranendonk (Van Kran- 
endonk 1961, and unpublished lecture notes). Since no relevant re- 
prints are reproduced in this volume, the derivation will be sketched 
in the present section. 

The vector potential of an electromagnetic wave, emitted by a 
classical oscillator of constant frequency, can be written as 

A(t) = Aq exp(io) t) 

where without loss of generality, Aq is normalized to |A | 2 = 1. If the 
frequency is a function of time, this equation is replaced by 

t 
A(t) = A exp [i /w(t')dt'] (19) 

o 

Assume for simplicity that the emitting oscillator moves in the x 
direction, with a velocity v(t) which is small compared to the light 
velocity c. The Doppler effect changes the frequency of the emitted 
wave 

w(t / ) = a) [l+v(t')/c] (20) 

Inserting (20) into (19) and integrating yields 

A = Aq exp(iw t) exp[iw x(t)/c] 
or 

A = Aq exp(iu> t) exp[ix(t)/X] (21) 

The essential features can now be discussed by letting the source of 
the electromagnetic wave execute a simple harmonic motion, with 
frequency to and amplitude Xq, 

x(t) = x sin m (22) 

The vector potential becomes 

A = Aq exp (iw t) exp (ix sin fit/*) (23) 

Such expressions are well known from the theory of frequency modu- 
lation in radio transmission. The spectrum, which originally just con- 
tained the carrier frequency co , splits up into lines of frequencies u> , 



18 THE MOSSBAUER EFFECT 

a) ±fi, cu ±2£2 .... This splitting is derived in the following way. Using 
the expansion 22 

oo 

exp (iy sin 0) = £] J n (y) exp (in0) (24) 

n=- °° 

Eq. (23) becomes 

oo 

A = Ao £ J n( x o/*) exp [i(w + nO)t] (25) 

n=-oo 

Equation (25) indeed describes an electromagnetic wave which is a 

superposition of partial waves with frequencies u> , a> ± £2, oo ± 2J2 

The amplitude of each wave is given by the Bessel function J n (x /x). 
The unshifted line can be identified with the Mossbauer line. Its inten- 
sity is given by 

f= |A(n = 0)| 2 = J 2 (x /X) (26) 

Since A is normalized, f directly yields the probability of emission 
of the unshifted component. 

So far, it has been assumed that the emitted wave is modulated by 
one frequency only. Applied to solids, this assumption corresponds 
to the Einstein model (Fig. 2- la). In order to use the previous dis- 
cussion for real solids, one must generalize (22), (23), and (25) to 
correspond to the correct frequency spectrum. The frequency Q, and 
the amplitude Xq are replaced by a sum over frequencies £2 m with 
amplitudes x m . One finds instead of (26) 



3N 

n Jj(x m /*) (27) 

m = l 



The number 3N of frequencies in a solid is extremely large. Each 
of the factors J 2 in (27) is only very slightly different from unity 
since the maximum amplitude x m of each individual frequency com- 
ponent is extremely small. Thus J can be expanded 

J (y) = l - (l/4)y 2 +-- 

and (27) can be written 



22 H. Jeffreys and B. S. Jeffreys, "Methods of Mathematical Phys- 
ics, " Cambridge University Press, New York, 1956, Sec. 21.10. 



THEORY 19 



In f = 2 E In J„ - 2 £ In [1 - (l/4)(x 2 m /* 2 )] 
m m 



--2 £ (1/4)04,/**) (28) 

m 

In this expression, In f is a function of the maximum excursions x m . 
Usually one introduces the mean- square deviation of the vibrating 
atom from its equilibrium position by the definition 

<x 2 > = (1/2)2 x 2 m (29) 

m 

and writes instead of (28) 

lnf e- -<x 2 >/X 2 

This equation is exact in the limit N— *> (Van Kranendonk 1961), so 
that the final result can be written 

i = exp(-<x 2 >/* 2 ) (30) 

Equation (30) leads to some interesting remarks: 

1. It is not surprising that the classical treatment yields an un- 
shifted line. The exact agreement between the classical expression 
(30) and the quantum mechanical result, as given in Sec. 2-5,Eq. (56), 
is unexpected, however. 

2. Equation (30) allows a simple physical interpretation. The con- 
tinuously emitted electromagnetic wave comes from a region of lin- 
ear dimensions <x 2 >. If this linear dimension increases beyond the 
wavelength fr= A/27T, pieces of the wave train emitted from different 
points in this region interfere destructively, and the fraction f of 
photons emitted without energy loss decreases rapidly. 

3. The condition for appreciable emission without energy loss, 
namely, that the amplitude of the emitting atom is small compared to 
the wavelength of the emitted photon, means that small spatial zero- 
point vibrations are essential for a large Mossbauer effect. The un- 
certainty relation then asserts that large zero-point momenta must 
be present. 

4. The wave train described by (23) is infinitely long and the cor- 
responding emission line is infinitely narrow. In decays with a mean 
life r, the lines possess a width T, Eq. (3), and the spectrum corre- 
sponding to (25) then is of the form shown in Fig. 2-2. From this 
spectrum it is obvious that the Mossbauer line is clearly recogniz- 
able only if the line width T is smaller than the separation tifi from 
the first satellite, i.e., if the nuclear lifetime t is larger than the 
characteristic lattice time I/O. This latter time is of the order of 
10" 13 sec. 



20 



THE MOSSBAUER EFFECT 




O) + 2Q 

Fig. 2-2 Spectrum of a classical electromagnetic wave of finite 
length emitted by an Einstein solid. 

5. Figure 2-2 gives a somewhat misleading impression since it is 
based on an Einstein solid without interaction among the various os- 
cillators. In an actual solid, whether described by an Einstein model 
with interaction (Van Kranendonk 1961) or by a Debye model, only the 
unshifted line shows the natural width T; all the satellites are broad- 
ened, overlap each other, and give rise to a continuum (Reprints on 
theory). 

6. The treatment outlined in this section can also be applied to ex- 
periments where the Mossbauer effect is observed with the source 
mounted on a crystal that oscillates with a frequency J2. Such an ex- 
periment is described in a reprint (Reprint Ruby and Bolef). 

2-4 THE PHYSICAL PICTURE 

Before sketching the quantum mechanical theory of the Mossbauer 
effect, a discussion of the physical picture may elucidate some of the 
aspects which can be lost easily in the formalism. Much of the dis- 
cussion is based on the unfailing war horse of physicists, the uncer- 
tainty principle. Part of the treatment follows lectures by Weisskopf 
(Weisskopf 1961). 

Three separate questions will be considered; momentum conserva- 
tion, energy conservation, and the time sequence and duration of 
events. 



2-41 Momentum Conservation 

Assume that the nucleus of an atom which is imbedded in a solid 
decays by gamma emission. If free, the nucleus would receive a re- 
coil momentum p and a recoil energy R, given by (1). How does the 
binding of the atom in the solid affect recoil momentum and recoil 
energy ? The answer to the first question is straightforward: The 



THEORY 21 

momentum is unchanged, but it is eventually taken up by the solid as 
a whole. In order to justify this statement, consider the two other 
possibilities, trans lational motion of the nucleus and phonons (lattice 
vibrations). The momentum cannot go into translational motion of the 
nucleus. The energy required to leave a lattice site is at least of the 
order of 10 ev; the energy available, however, never exceeds a few 
tenths of an ev. (Even if the recoil were larger, the nucleus would 
finally come to rest and transfer its momentum to the solid.) Lattice 
vibrations, on the other hand, cannot take up momentum. They can be 
represented as standing waves or as the sum of running waves. To 
each wave with its momentum pointing in one direction will be a cor- 
responding one with its momentum pointing in the opposite direction. 
The expectation value of the momentum for lattice vibrations van- 
ishes. (Even if the recoiling nucleus initially starts a Shockwave that 
travels through the lattice, the Shockwave will finally be damped out, 
and one is led back to the equilibrium situation discussed here.) The 
momentum hence must go into translational motion of the entire crys- 
tal. If the crystal is glued to a larger body, for instance the earth, 
this larger body takes up the momentum. The momentum conservation 
is taken care of and one can now turn his attention to the energy con- 
servation. (Incidentally, the nearly complete separation of the energy 
transfer from the momentum transfer which occurs in the Mossbauer 
effect appears also in many classical problems, such as when one 
shoots a bullet into a very heavy pendulum.) 

2-42 Energy Conservation 

The discussion of the energy conservation is more complicated, 
since the transition energy can be shared among the gamma ray, the 
individual atom, lattice vibrations, and the solid as a whole. Two of 
these four parts can be dispensed with quickly. The individual atom 
does not leave its lattice site (see Sec. 2-41) and hence cannot acquire 
translational energy. The energy that goes into motion of the entire 
solid is extremely small and will be neglected. The transition energy, 
for all practical purposes, is thus shared between the gamma ray and 
the phonons. A Mossbauer transition occurs if the state of the lattice 
remains unchanged, and the gamma ray gets the entire transition 
energy. 

The fact that transitions can occur in which the lattice remains in 
its initial state and the gamma ray receives the full energy follows 
from the quantum mechanical treatment (Sec. 2-5). A naive picture 
can be given which makes this result plausible for the Einstein solid, 
as well as for a solid with a continuously distributed frequency spec- 
trum. For the Einstein solid, the problem is very simple. The smal- 
lest amount of energy that can be given to the solid is equal to 
Eg =-ncoE = keg- ^ tne ener SY R U- e »> the recoil energy of zfree 
nucleus) is small compared to this excitation energy, the probability 



22 THE MOSSBAUER EFFECT 

of emission of a phonon will be small, the lattice will not be excited 
and the gamma ray will escape with the full transition energy. The 
calculation (Sec. 2-5) shows indeed that the probability f for a tran- 
sition without energy loss is given by 

f = exp(-R/k0 E ) (31) 

For the Debye solid, the situation is more complicated. Assume 
first that the decay proceeds in such a way that a lattice vibration of 
the shortest possible wavelength, i.e., the maximum energy E^ = "hw D 
= k0j), is excited. The wavelength of this shortest lattice wave is 
A.«2d, where d is the lattice constant. In a continuum theory, such 
as the Debye theory, the corresponding energy is 

E D f fiw D = flu/* « 2irftu/2d (32) 

where u is the sound velocity in the solid. If this particular lattice 
wave were the only one that could be excited, one would be in the same 
case as before and (31) would hold, with E replaced by 0r> How- 
ever, lattice modes with longer wavelength and hence smaller energy 
exist, and one would expect that these vibrations can be excited easily, 
thus making a Mossbauer effect impossible. The fact that these modes 
with smaller energy cannot be excited too easily can be seen as fol- 
lows. The highest mode corresponds approximately to a situation 
where two adjacent atoms move out of phase. Such a wave can be ex- 
cited most efficiently if the decaying atom is assumed to be free, re- 
ceives its full share R of the recoil energy, and then bumps into a 
neighboring atom. A simple mechanical analog is the well-known dem- 
onstration experiment in which a number of spheres hang from a frame 
and touch each other. Lifting the outermost sphere and letting it bump 
into the row excites a wave which travels through the chain and causes 
one sphere at the other end to jump off. In order to excite longer 
waves in this mechanical model, one lifts N spheres at one end, re- 
leases them, and N will bounce off at the other end. Similarly, a 
longer wave can be excited most efficiently in the solid if initially N 
atoms move together. The wavelength then is approximately 2Nd, and 
the energy of this wave is, according to (32), about Erj/N. However, 
in order most efficiently to excite this longer wave, N atoms must 
move together at the onset, and the recoil energy must be transferred 
to these N atoms "simultaneously." The decaying system is no 
longer one atom alone but the N atoms together. The mass of this 
system is NM, and the recoil energy given to it is R/N. So, even 
though this wave requires only Ep/N as excitation energy, only R/N 
is available. 

The proper calculation (see Reprints on theory) justifies these con- 
siderations and shows that the fraction f of transitions without change 



THEORY 23 

in the lattice states is given by an expression similar to (31), but with 
E replaced by (2/3)6d: 

f = exp(- 3R/2k0 D ) (33) 

Equations (31) and (33) are valid only at zero absolute temperature, 
where all lattice oscillators are in their ground state. At finite tem- 
peratures, some of the oscillators are excited and transitions with in- 
duced emission of phonons become possible. These transitions con- 
tribute an additional term in the exponents of (31) and (33). 

The foregoing discussion is obviously oversimplified and, in addi- 
tion, treats a quantum mechanical problem in a classical way. How- 
ever the essence is correct. Even though the energies characteristic 
for the solid are much smaller than the nuclear transition energy, and 
the nuclear decay occurs in one nucleus only, the entire crystal must 
be considered as the quantum mechanical system in which the decay 
occurs. Any statement according to which one can separate the decay 
into a first step, in which the nucleus decays, and a second step, in 
which the recoil energy is, or is not, given to the solid, is misleading. 
The process is indivisible and if , by a measurement , one separates 
the two steps, the Mossbauer effect is destroyed. These statements 
are justified in the next section. 

2-43 Time Considerations 

Before discussing the times involved in the Mossbauer effect, a 
few words about the uncertainty relation between energy and time are 
in order. It has been pointed out quite early, for instance by Pauli, 23 
that this uncertainty relation has two different physical contents. The 
one of interest here refers to a measurement: If a system has two 
states, E n and E m , then any measurement to decide whether the sys- 
tem is in state n or in state m requires at least a time T given by 

T«V(E n -E m ) (34) 

This careful definition avoids some of the difficulties that have led to 
many discussions (for instance, see Ref. 24). 

Consider, as a special case, the 14.4-kev transition from the first 
excited state to the ground state of Fe 57 . According to (34), the min- 
imum time necessary to decide whether a Fe 57 nucleus is in its 
ground state or its first excited state is 4xl0" 20 sec. The question 
then arises as to whether this time is characteristic for this transi- 
tion so that one can say that the individual decay always occurs faster 
than 4 x 10' 20 sec. 



23 W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik, 
Handbuch der Physik," Springer, Berlin, 1933, Vol. 24:1, p. 146. 

24 Y. Aharonov and D. Bohm, Phys. Rev., 122, 1649(1961). 



24 THE MOSSBAUER EFFECT 

To answer this question, a gedanken experiment can be performed 
in which the nucleus is observed at intervals of 4x 10" 20 sec, and thus 
the moment of decay is determined within this time. Has one then 
shown that a transition time of 4 x 10" 20 sec or less exists, or has the 
measurement forced the nucleus to decay within this short time ? As 
Pauli's statement in Sec. 2-1 shows, this question had obviously been 
discussed very thoroughly when quantum mechanics was new, and 
the second alternative was found to be correct. One of the beautiful 
aspects of the Mossbauer effect is that it gives a direct experimental 
confirmation of these quantum mechanical ideas. With Fe 57 one can 
indeed perform an experiment similar to the ' 'thought' ' experiment 
described above, if only with intervals considerably longer than 10" 19 
sec but still much shorter than 10" 7 sec, the lifetime of the first ex- 
cited state of Fe 57 (see Reprint Lynch 1960 and Sec. 5-6). The spec- 
trum of gamma rays emitted in such an experiment no longer dis- 
plays the natural line width but is considerably broadened. To a good 
approximation, this broadening is given by (3), with r now standing 
for the interval of measurement. This experimental result agrees 
with the quantum mechanical calculations (Harris 1961) and also with 
the classical considerations 6 : If the emitted wave train is shortened 
in time, its energy distribution, which is given by the Fourier trans- 
form of the time distribution, must be wider. In the thought experi- 
ment above, with time intervals of 4 x 10" 20 sec one would find out 
whether the nucleus has decayed or not, but the gamma ray would 
show a width of about 15 kev instead of the natural line width of about 
10" 8 ev. Any Mossbauer effect would clearly be impossible. These 
considerations show that it is meaningless to talk about a ''transition 
time" or about an "instantaneous transition.' ' The times that enter 
are those characteristic for the measurement. 

After these preliminary remarks, the discussion of the Mossbauer 
effect is straightforward. If one wants to ascertain whether or not a 
transition with energy transfer AE to the lattice has occurred, one 
must measure the energy of the photon or of the solid to within the 
energy AE. The time required for this measurement is given by (34), 

T>1\/AE = ^r/u (35) 

where 2ttX is the wavelength belonging to a phonon with energy AE 
and velocity u. During the time T, the disturbance caused by the de- 
cay travels a distance L, given by 

L = Tu > X (36) 

Thus, if one wants to determine whether or not an amount of energy 
AE has been transferred to the lattice, one is forced to look at re- 
gions of the solid of linear dimensions X or larger. This argument 
justifies the statements made in Sec. 2-42. 



THEORY 25 

It is impossible to decide by a measurement whether the original 
recoil has been imparted to the decaying nucleus only or has been 
shared among two, three, or more atoms within a region of linear di- 
mensions X. 

If no particular time is selected by an external apparatus, the mean 
life t of the decaying state will play the role of the measuring inter- 
val. During one mean life, the disturbance from the decay will travel 
for a distance tu. The question then arises whether a measurement 
can decide that a part of the entire solid, of linear dimensions tu, 
will begin to move first. It is easy to see, however, that the recoil 
energy of such a crystallite cannot be measured, because zero-point 
vibrations will completely mask any recoil effect (Weisskopf 1961). 

There remains one last question. How do the parts of the solid far 
away from the decaying nucleus know how and when to move ? Is it 
possible to decide whether or not the signal travels through the crys- 
tal as a shock wave with the velocity of sound u ? Again the uncertainty 
relation prevents such a detailed investigation. The recoil energy 
given to the entire solid is approximately 

R solid = R ( M atom/M so iid) * R( d /D) 3 (37) 

where R is the recoil energy of the free atom, d is the lattice pa- 
rameter, and D the linear dimension of the solid. The recoil energy 
is of the order of the Debye energy, Eq. (32): 

R«liu/d 

The time required to measure the recoil energy of the entire solid is, 
according to (34), given by 

T s * VRsolid « *D 3 /Rd 3 « D 3 /ud 2 

The time t for a signal with the sound velocity u to travel the linear 
dimension D is t = D/u. The ratio of the two times thus becomes ap- 
proximately 

T/t«(D/d) 2 (38) 

The time required for a measurement of the recoil energy of the en- 
tire solid is very much larger than the time needed for a signal to 
travel through the crystal. By the time the measurement of the recoil 
energy is finished, the crystal moves with uniform velocity (if not 
glued to some material), and it is impossible to tell how it received 
its travel orders ! 



26 THE MOSSBAUER EFFECT 



2-5 SKETCH OF THE THEORY 

In nearly every theory there exist steps that are omitted in the 
theoretical papers and not treated in the textbooks. These steps are 
obviously designed to keep the experimental physicists in their place. 
The theory of the Mossbauer effect makes no exception; the equation 
from which all later results are derived [(40) below] is usually writ- 
ten down without detailed derivation. It is the purpose of the present 
section to supply the steps missing in most papers. 

The probability for emission or absorption of a gamma ray from a 
nucleus embedded in a solid, with the lattice simultaneously undergo- 
ing a transition from one state to another, is usually calculated by 
dispersion theory, as for example described in Heitler. 6 The relevant 
formula will not be needed in the present section, but it can be found 
as Eq. (8) in Lamb's paper (Reprint Lamb 1939). f Here the essential 
fact used is that the transition probability for a given transition is 
proportional to the square of the matrix element connecting the two 
states involved in the transition (golden rule). In particular, the prob- 
ability for a transition in which the nucleus decays from the excited 
state Ni to the ground state Nf , while simultaneously the lattice goes 
from its initial state L^ to its final state Lf , is 

W(Ni-N f , Li— Lf) = const. |<f |H int |i> | 2 (39) 

where |i> and <f | denote the initial and final state of the entire sys- 
tem, including the lattice, and H^ is the interaction Hamiltonian re- 
sponsible for this decay. The energy that can be transferred to the 
lattice during this transition is very small compared to the gamma- 
ray energy. The dependence of the density of final states p(E) on the 
energy transfer to the lattice is hence very small, p(E) is assumed 
to be constant, and it is absorbed in the constant in (39). 

For calculations, an explicit form of the transition matrix element 
is needed. Lamb states that, because of the short range of nuclear 
forces and the corresponding independence of the motion of the crys- 
tal of the center of momentum (cm.) and the internal degrees of 
freedom of the nucleus, it can be factored into a nuclear and a lat- 
tice matrix element: 

<f|H int |i> = <L f |eik-X| Li><N> ( 40 ) 

Here the nuclear matrix element <N> depends only on nuclear prop- 
erties, k = p/h is the wavevector of the emitted gamma ray, and X 
is the coordinate vector of the cm. of the decaying nucleus. 



t A derivation based on time- dependent perturbation theory is given 
in Appendix A of a paper by Petzold (Petzold 1961). Compare, how- 
ever, the physical interpretation of Petzold with the remarks made 
here in Sec. 2-4. 



THEORY 27 

<Lf |exp(ik*X)|Li> is the matrix element for transfer of a momen- 
tum -hk to the lattice through the atom of the decaying nucleus with 
the lattice going from the state L i to Lf . 

Lipkin (Reprint Lipkin 1960) shows that the form Eq. (40) of the 
matrix element is determined completely by the requirements of 
translational and Galilean invar iance. A more detailed justification 
of (40) may, however, be desired by many readers since it forms the 
basis for all the later calculations and arguments. Such a justifica- 
tion, in a very pedestrian way, will be given here. 

The Hamiltonian of a charged particle, moving with a momentum p 
in an electromagnetic field given by a vector potential A, contains the 
term [p - (e/c)A] 2 . This term leads to the nonrelativistic interaction 
Hamiltonian Hi n t = const (p • A - A-p). After expanding A in plane 
waves, one finds the well-known expression 6 ' 25 

<f|H int |i> = const. <f|exp(ik-x)p A |i> (41) 

where the gradient operator pa must be applied in the direction of 
the polarization vector of the electromagnetic wave k, and where x 
is the coordinate vector of the decaying nucleon. Equation (41) corre- 
sponds to a single-particle description of the nucleus; if the decay oc- 
curs through many nucleons in the same nucleus, an appropriate sum 
over these nucleons must be introduced in (41). For simplicity, this 
sum is omitted. 

The short range and the strength of the nuclear forces now permits 
the use of the approximation that the nuclear decay is not influenced 
by the state of the lattice and that the lattice condition does not depend 
on the nuclear state. The state functions |i> and <f | can then be writ- 
ten as products |N^>|Li> and <Lf|<Nf| of nuclear- state functions 
|Nj> and <Nf| and lattice- state functions |L^> and <Lf|, with 
<Lf|Li> = 6fi and <Nf|Ni> = 5fj. In addition, one introduces inter- 
nal nuclear coordinates p by writing 

x = X + p (42) 

where X is the coordinate vector of the cm. of the decaying nucleus 

(Fig. 2-3). 

The momentum operator also splits up into a sum 

PA = PX + Pp ( 43 > 

where pp acts on the internal nuclear coordinates and px on the 
cm. coordinates of the nucleus. After introducing (42) and (43) into 
(41), the matrix element can be separated into two parts, each con- 
sisting of a product of a nuclear and a lattice matrix element: 



25 L. I. Schiff, "Quantum Mechanics,'' McGraw-Hill, New York, 
1955, Sees. 23 and 35. R.P.Feynman, "Quantum Electrodynamics,' 
W. A. Benjamin, Inc., New York, 1961, p. 8. 



28 



THE MOSSBAUER EFFECT 



radiating nucleon 




decaying nucleus 



cm. 



origin 



Fig. 2-3 Coordinates used in the evaluation of the transition matrix 

element. The origin is fixed at the cm. of the entire crystal. 



<f|H m t|i> = const. 



<Nf|e ik ' P p p |N i ><L f |e ik,X |L i > 



+ <N f |e ik * p |Ni><Lf|e lk,X p x |Li>} 



(44) 



To reduce (44) to (40) one must show that the first term in (44) is 
much larger than the second one. The following arguments are only 
qualitative, but they yield the right order of magnitude. Using the 
closure 25 ' 26 



L |n'Xn'| = 1 
n' 



(45) 



where the summation extends over all intermediate states, the ratio 
of the lattice matrix elements in the two terms of (44) can be written 



<Lf|e ' pxjLj, 
<L f [e ik ' X |L i > 



g<L t |e tk,X |L'XL'|i> x |L 1 > 
<Lf|e ik - X |Li> 



The momentum "hk transferred to the lattice during the nuclear decay 
is much larger than typical momenta components in the lattice. It is 
therefore to be expected, even without detailed calculation, that the 
terms <Lf |exp(ik*X)|L'> for allowed intermediate states L' are of 
the same order of magnitude as the term <Lf|exp(ik«X)|Li> . They 
can then be taken out from under the sum sign, leading to 



r L « L <L'|px|Li> =<p L > 



(46) 



26 P. A. M. Dirac, "The Principles of Quantum Mechanics, " Oxford 
University Press, New York, 1958, 4th ed., p. 63. 



THEORY 29 

where <Pl> denotes an average over lattice momentum components. 
A similar reasoning is valid for the nuclear matrix elements, where 
the momentum transfer 1ik is much smaller than typical nuclear mo- 
mentum components. The corresponding ratio of the nuclear matrix 
elements appearing in (44) becomes 

r N « L<N'|Pp|Ni> =<p N > (47) 

N' 

The ratio of the first to the second term in (44) thus is 

r N/ r L ^<pn>/<p l > < 48 ) 

or of the order of 10 5 or larger. A second way of estimating the ratio 
(48) can be obtained by first calculating rjg, using a multipole expan- 
sion, exp(ik-p) = 1 + ik-p + ••• . One finds for the first nonvanishing 
terms of the relevant matrix elements: <Nf |exp(ik*p)|Ni> 
« ik<N f |p|Ni>, and <N f |exp(ik-p)p p |Ni> « <Nf |p p |Ni> . The last 
expression can be evaluated further by assuming aHamiltonian of the 
form H = (p 2 /2M) + V(p). One then easily calculates a commutator 
[H,p] = (m/M)p, or p = (-iM/ti)[H,p]. Hence <N f |p|Ni> = (-iM/n)x 
<N f |Hp - pH|Ni> = (+iM/h)(Ei - Ef)<N f |p|Ni> = +iMkc<N f |p|Ni>. 
The desired ratio r^ becomes r N = -Mc ^<pjsp> > * n agreement with 
the crude arguments leading to (47). To estimate r^ one first re- 
marks that the highest momenta occurring in the lattice are of the or- 
der of H/d and that the matrix element <Lf|exp (ik*X)px|Li> thus 
cannot be larger than ti/d. The matrix element < Lf|exp (ik*X)|Lj> , 
however, must be larger than a certain minimum value, say 10" 2 , for 
the Mossbauer effect to be observable [compare (49) below]. Thus 
rL « lC^-h/d and r^/r^, =* Mcd/10 2 ti ^10 6 . These arguments show 
that (40) can safely be used. 

Using (40), the calculation of the fraction f of gamma rays emitted 
without energy loss to the lattice (Li — Li) is now straightforward: 

f= _KLi|e ik ' X |Li>| 2 



L |<L f |e ik,X |Li>; 
Lf 



The denominator is easily shown to be one with the help of (45), 
Il<L f |e lk - X |L i >p=i:<L i |e- ik - X |L t ><L £ |e ik - X |L i > 



Lf Lf 



<Li|e e |Li> = 1 



Thus one finds 



30 THE MOSSBAUER EFFECT 



f=|<L i |e ik,X |L i >| 2 (49) 

Equation (49) has been the starting point for most calculations of 
the fraction f of gamma rays emitted or absorbed without energy loss. 
A few examples best illustrate its use. 

2-51 The Einstein Solid 

The simplest and most pedestrian application is to the Einstein 
solid. The ground- state wave function for a linear harmonic oscilla- 
tor of mass M and angular frequency u) is given by 

* (x) = iWS e ~^^ 2n (50) 

Inserting this wave function into (49) and using the fact that one is 
dealing with a one-dimensional problem, i.e., that k-X — kX, yields 
after integration 

f = exp (--h 2 k 2 /2Mfta>) = exp (-R/nu) E ) (51) 

This result agrees with (31), which was quoted earlier without proof. 



2-52 The Debye Solid 

For the Debye solid, the procedure is somewhat more complicated, 
because the individual atoms do not all have the same frequency. One 
introduces normal coordinates and then determines the probability f . 
This straightforward calculation is performed in a number of papers 
(Reprints Mossbauer, Lipkin, Visscher). The result can be written 

f = e' 2w (52) 



where 



w - R 



ke D 



1 /T_\ 2 e/T xdx 
4 W / x . 



e 



(53) 



This result agrees for T = with (33). It must be pointed out that 
it is only valid if the concept of Debye temperature is applicable. In 
many instances, even for crystals consisting of only one kind of atom, 
the Debye theory cannot be used. 18 When estimating f with the help of 
Eqs. (52) and (53), these restrictions should be kept in mind. For 
obtaining crude estimates of expected effects, the Debye approximation 
is, however, very useful. In Sec. 3-6, data helpful for such estimates 
are collected. 



THEORY 31 

2-53 The Debye- Waller Factor 

At this point it is worth remembering that recoilless processes 
have not been invented by nuclear physicists. The scattering of X 
rays without loss of energy to the lattice has been used as a standard 
tool for many years. In studies of Bragg reflection it was found that 
the temperature had a strong influence on the intensity of the lines. 
Many authors, particularly Debye and Waller, investigated this effect 
and found that the intensity as a function of temperature could be ex- 
pressed as 27 " 31 



I=I e 



-2W 

W = 2<u|> sin 2 <p/% 2 (54) 

where <p is the Bragg angle, 2-nX is the X-ray wavelength, and<u|> 
is the mean- square deviation of the component of displacement of the 
atoms along the direction z which is perpendicular to the reflecting 
planes. This Debye-Waller factor exp(-2W) has many similarities 
with the Lamb-Mdssbauer factor f ; it is interesting to compare these 
and also note the differences. 

The main difference lies in the times involved. X-ray scattering 
is "fast," the characteristic time involved is much shorter than the 
characteristic lattice time. In contrast, the emission and the scatter- 
ing of gamma rays in the Mossbauer effect is "slow," the relevant 
time is comparable to, or longer than, the characteristic lattice time. 
These differences are discussed in papers by Tzara (Reprint Tzara 
1961) and Trammell (Trammell 1961), and these should be consulted 
for further details. 

The essential similarity lies in the appearance of the mean-square 
deviation of the radiating or scattering atoms from its equilibrium 
position. For crystals with harmonic lattice forces, it can be shown 
that the expression (49) can be transformed to 

f = exp(-<Li|(k.X) 2 |Li» (55) 

or 



27 P. P. Ewald, "Handbuch der Physik," Springer, Berlin, 1933, 
Vol. 23:2, p. 307. 

28 F. C. Blake, Revs. Modern Phys., 5, 169 (1933). 

29 A. H. Compton and S. K. Allison, "X-Rays in Theory and Exper 
iment," Van Nostrand, Princeton, N. J., 1935, p. 437. 

30 M. von Laue, "Ro'ntgenstrahlinterferenzen," Akademische Ver- 
lagsgesellschaft, Leipzig, 1948, pp. 204, 242. 

31 R. W. James, "Optical Principles of the Diffraction of X-Rays,' 
Bell, London, 1948. 



32 THE MOSSBAUER EFFECT 

f = exp(-<L i |Xk|L i >/* 2 ) (56) 

Here X^ is the component of the coordinate vector X in the direction 
of the emitted photon. Equation (56) is derived in Petzold's paper 
(Petzold 1961) and the steps leading from (49) to (55) can be found in 
a paper by Van Hove. 32 ' 33 Incidentally, (56) agrees with the classically 
derived Eq. (30) and the remarks made there apply also to (56). 

2-54 Connection with the Probability Density 34 

If one denotes with p(X) the probability of finding the radiating nu- 
cleus at a distance X from its equilibrium position, one can write 

f= |/e ik ' X p(X)d 3 X| 2 (57) 

Equation (56) is obtained from (49) by integrating the latter matrix 
element over all lattice variables except X. It shows that f is the 
square of the Fourier transform of the density p(X). 

2-55 Further Calculations 

In the back of this volume a number of reprints are collected which 
further develop the theory of the Mossbauer effect. Mossbauer's pa- 
per, in which he adapts Lamb's theory, is reprinted in German and 
the section on theory is translated. In Visscher's paper, Mossbauer's 
results are derived in a more modern way and applications to a study 
of lattice vibrations are indicated. Lipkin's paper deals with simple 
sum rules which give a deeper physical understanding of recoilless 
processes. Inglis, in a paper not printed elsewhere, discusses the 
theory of the Mossbauer effect, first in a very simple one-dimensional 
case and then generalizes to three dimensions. Singwi and Sjolander 
use the space- time self- correlation function of Van Hove 33 to arrive 
at very elegant and general results. This approach is probably the 
one that is most useful and versatile for applications. Tzara includes 
the interference with other processes in his paper and also shows 
the connection with X-ray scattering. For all further details con- 
cerning theory, these papers should be consulted. 



32 L. van Hove, Phys. Rev., 95, 249 (1954). Contained in Ref. 33. 

33 L. van Hove, N. M. Hugenholtz, and L. P. Howland, ''Problems 
in the Quantum Theory of Many- Particle Systems," W. A. Benjamin, 
Inc., New York, 1961. 

34 H. J. Lipkin, unpublished notes, 1961. 




EXPERIMENTAL APPARATUS 
AND PROBLEMS 



3-1 SURVEY 



In contrast with experiments in high-energy physics, investiga- 
tions involving the Mossbauer effect are quite simple, easy to under- 
stand, and inexpensive. Still, difficulties exist, only a few of which 
have been solved completely. Consider, for instance, the vibration 
problem in Zn 67 . This isotope displays a very narrow resonance 
with a width of about 10" 10 ev. This width corresponds to a Doppler 
velocity of about 10" 5 cm sec" 1 , or, as the Los Alamos group puts it, 
a velocity slightly faster than the one with which fingernails grow. 
If hum exists in a velocity drive used with Zn 67 , an amplitude of a 
few times 10" 8 cm can be sufficient to destroy the resonance! 

A basic setup for a Mossbauer experiment and a typical result 
are sketched in Fig. 3-1. The source is moved with a velocity v 
with respect to the absorber, (in many experiments, it is more con- 
venient to move the absorber.) The gamma ray then suffers a Dop- 
pler shift AE, 

AE = (v/c)E (58) 



where E is the gamma-ray energy. The velocity v is defined as 
positive if the source moves toward the absorber. The intensity 
I in the detector is determined as a function of the velocity v. 

At large velocities, no resonance absorption occurs. The velocity 
at which resonance absorption becomes vanishingly small depends 
on the line width and on the line splitting. In each experiment one 
usually selects a "safe" velocity and denotes it with v^. (Visitors 
to our lab are sometimes startled at seeing signs such as "°° = 
1.3 cm/sec") One plots l(v) = I - B, where B is the background 

33 



34 



THE MOSSBAUER EFFECT 



source and 
velocity drive 



absorber 



counter 




X\\\\\\\\\\^\\\\\\\\\\\\\\\\\\\\\\ 



A 



k 



W v) 



"\/~ 



■*- V 



emission line absorption line 



res 

spectrum observed as a 

function of the 

relative velocity v 

(thin source and thin absorber) 



Fig. 3-1 Basic setup, emission and absorption lines, and 
velocity spectrum in a Mossbauer transmission 
experiment. 

More convenient is the normalized velocity spectrum l(v)/l(°°), or 
the deviation from nonresonant absorption, 



€(v) 



I(°°) - I(v) 
~1R 



(59) 



Figure 3-1 displays only the simplest case, single line source and 
single line absorber, both showing the natural line width. The veloc- 
ity spectrum can be much more complicated in actual experiments. 
The emission and the absorption lines can show different widths, T e 
and T a , and they can be split into components. The magnitude of the 
splitting and the number of components can be different for source 
and absorber. There can also be a shift between the centers of 
emission and absorption lines. From the experimentally determined 
velocity spectrum for a given source- absorber combination, one tries 
to determine one or more of the following quantities: Lamb- Moss- 
bauer factor, line shape, line splitting, and line shift. These quan- 
tities will be discussed in the Sees. 3-11 to 3-14. 



3-11 The Lamb- Mossbauer Factor 

The Debye-Waller factor exp(-2W) for X-ray scattering is ex- 
pressed by (54). The analogous quantities in the Mossbauer effect, 
sometimes called the Lamb- Mossbauer factors, are f, the fraction 



APPARATUS AND PROBLEMS 35 

of gamma rays emitted without energy loss, and f , the correspond- 
ing quantity for absorption. The factors f and f give information 
similar to that contained in exp(-2W). Since exp(-2W) is often dif- 
ficult to determine accurately by conventional X-ray techniques, it 
is desirable to measure f and f as a function of temperature for as 
many substances as possible. 

The usual methods for determining absorption coefficients are 
not applicable for two reasons: (l) The absorption by a resonant ab- 
sorber of a gamma- ray beam containing a fraction f of gamma rays 
emitted without energy loss is not exponential. The deviation is due 
to the fact that the absorption cross section varies sharply with en- 
ergy; the energy spectrum of the gamma rays is thus a function of 
the distance in the absorber. (2) The absorption is not only deter- 
mined by f but also by f. Both f and f must be extracted from 
the experiments. 

The determination of f and f is simple as long as the self- ab- 
sorption in the source is small and the emission and absorption lines 
are unsplit and have Lorentz shapes of identical widths. This case 
was first treated by the Los Alamos group (Craig et al. 1959) and 
the relevant equations are given in the paper by Margulies and 
Ehrman (Reprint Margulies 1961). More information can be found 
in the review by Cotton (Cotton 1960). 

If the emission or the absorption line or both are split, but still 
display identical line widths, an analysis similar to the one described 
in the Los Alamos paper can be performed (see Margulies 1962). An 
analysis along the same lines can also be done if the lines show a 
Gaussian shape but still have identical widths. In general, however, 
the experimentally observed shape will be neither Lorentzian nor 
Gaussian and the data evaluation will hence be more difficult. The 
evaluation is also considerably more difficult if emission and ab- 
sorption lines do not have the same widths. 

In many cases, the area method is useful. Here one considers 
the area under the absorption curve and derives a value of ff'. Often 
one may have enough knowledge to separate this product. This 
method, which was first used by the Argonne group (Hanna et al. 1960), 
is described in detail in a publication by Shirley, Kaplan, and Axel 
(Shirley 1961). 

3-12 Line Shape 

Only in the ideal case do emission and absorption lines display 
the natural line shape. In actual experiments, the lines are widened 
by finite source and absorber thickness and usually also by the finite 
velocity resolution of the apparatus (Sees. 3-4 and 3-5). Further- 
more, solid-state effects such as internal fields, imperfections, and 
impurities broaden the line and change its shape (see Chapter 6). 



36 THE MOSSBAUER EFFECT 

Experimentally, one wants to find the true shape and width of the 
emission and the absorption lines, extrapolated to zero source and 
absorber thickness. If the lifetime of the excited nuclear state is 
known from independent experiments (electronic delay measure- 
ments), one can compare the observed line width with the theoret- 
ically expected one. 

3-13 Line Splitting 

Extranuclear fields can split the emission and the absorption lines 
into components. These splittings, caused by magnetic and electric 
hyper fine interactions, are extremely important for the investigation 
of solid-state properties. Many typical examples are quoted in Chap- 
ter 6. One tries experimentally to determine the number of com- 
ponents and their separation from the velocity spectrum. It is clear 
that this requires a sufficiently high velocity resolution and a knowl- 
edge of the absolute velocity scale. 

3-14 Shifts 

The emission and the absorption lines are only centered at the 
same energy if the corresponding nuclei are in very similar environ- 
ments. This is very often not the case, and the two spectra are dis- 
placed one with respect to the other. This shift is deduced from the 
velocity spectrum. Shifts can result from temperature differences 
between emitter and absorber (Sec. 5-4) and from isomeric effects 
(Sec. 4-4). 

One more remark needs to be added about Fig. 3-1. The classical 
investigations on optical and conventional nuclear resonance fluores- 
cence were all performed as scattering experiments. In contrast, 
nearly all the work on the Mossbauer effect is done in transmission. 
The main reasons for this difference are the relatively large effects 
(between about 1 and 50 per cent in most cases) and the large inten- 
sities and favorable geometry obtainable in transmission. The inten- 
sity disadvantage of scattering is increased by the strong conversion 
of low- energy gamma rays; only a fraction l/(l + a) of the absorbed 
gamma rays is reemitted [Eqs. (9), (10), and (13)]. Some scattering 
experiments have nevertheless been performed, either by observing 
the gamma rays directly (Reprint Barloutaud 1960), by measuring 
the X rays that follow the internal conversion process (Frauenfelder 
1961), or by detecting the conversion electrons (Kankeleit 1961; 
Mitrofanov 1961). 

3-2 ISOTOPES 

Mossbauer discovered the recoilless gamma-ray emission with 
the 129-kev gamma ray in Ir 191 . If this isotope were the only one 
displaying such an effect, experiments would indeed be very difficult, 



APPARATUS AND PROBLEMS 37 



the applications would be limited, and interest would not be very great. 
Fortunately, however, there exist at least 15 nuclides in which Moss- 
bauer effect has been observed. There is a good chance that more 
will be found, and the number of possible experiments is extremely 
large. 

In Table 3-1, properties of nuclides that have been used for Moss- 
bauer experiments or which show promise for exhibiting such an ef- 
fect are collected. 35 Very likely this table is not complete and it is 
also possible that it contains errors. Before embarking on any ex- 
periments based on this table, it is wise to reevaluate all critical 
data. 

The table was originally compiled by D. Nagle (Los Alamos 
Scientific Laboratory) for the first Mossbauer conference held at 
the University of Illinois in June 1960. It was modified and checked 
by G. DePasquali, R. Morrison, and D. Pipkorn (University of Illinois) 
Similar tables can be found in the review by Cotton and that by Belo- 
zerskii and Nemilov. 

The entries in the table are mostly self-explanatory, but the fol- 
lowing notes may be of some assistance: 

Powers of ten are denoted as follows: 1.0 x 10" 7 — 1.0(-7). 

a = abundance of stable element, % 

Q = ratio of gamma- ray energy Ey to line width r = ft In 2/Ty 

a = total internal conversion coefficient (K implies aj£, 
L implies a^J 

R = recoil energy of free nucleus in units of 10" 2 ev 

o' = Mossbauer absorption cross section in units of 10" 19 cm 2 , 
calculated with the listed values of the conversion coef- 
ficient and assuming natural line width. In the case of an 
unknown conversion coefficient the cross section has been 
calculated by assuming a = 0. Once a is known, the cor- 
rect cross section can be found by multiplying the value 
in the table by l/(l + a) 

X = an effect has been observed 



Table 3-1 

Nuclides of Interest in Mossbauer Experiments 



Stable 




Ey, 


T%, 








nuclide 


a, % 


kev 


sec 


Q a, e/y 


R, 10 ~ 2 ev 


ff' 0f 10 ~ 19 cm 2 Effect 


Fe" 


2.17 


14.4 


1.0(-7) 


3.2(12) 15 


0.19 


15 X 


Ni" 


1.25 


71 


5.2(-9) 


8.1(11) K 0.11 


4.4 


6.6 X 


Zn 67 


4.11 


93 


9.4(-6) 


1.9(15) K 0.63 


6.9 


1.2 X 


Ge" 


7.67 


13.5 


3.1(-6) 


1.4(14) 3600 


0.13 


0.030 

(continued) 

as been taken 


35 Most of the information contained in ' 


Table 3-1 h 



38 



THE MOSSBAUER EFFECT 



Table 3-1 (continued) 



Stable 
nuclide 



Ey 
kev 



T l / 2 
sec 



ae/y R, 10 -2 ev 



10" 



Effect 



Kr 83 


11.55 


9.3 


< K-7) 




10 


0.055 


Ru 99 


12.7 


89 








4.3 


Ru 101 


17.0 


127 


1.4(-9) 


3.9(11) 


K 0.4 


8.5 


Ag 107 


51.35 


93 


44.3 


9.0(21) 


16 


4.3 


Ag 109 


48.65 


88 


39.2 


7.6(21) 


14 


3.8 


Sn 117 


7.57 


161 






K0.13 


12.0 


Sn 119 


8.58 


24 


1.9(-8) 


1.0(12) 


7.3 


0.26 


Sb 123 


42.75 


161 








11.0 


Te 123 


0.87 


159 


1.9(-10) 


6.6(10) 


K 0.17 


11.0 


Te 125 


6.99 


35 


1.6(-9) 


1.2(11) 


K 12 


0.52 


I 127 


100 


59 






1.9 


1.5 


Xe 129 


26.44 


40 








0.67 


Xe 131 


21.18 


80 


4.8(-10) 


8.4(10) 


K 1.73 


2.6 


Cs 133 


100 


81 


6.0(-9) 


1.1(12) 


K 1.5 


2.7 


La 139 


99.9 


163 


1.5(-9) 


5.4(11) 


K 0.22 


10.0 


Nd 14S 


8.29 


67 


3.3(-8) 


4.8(12) 


K 3.3 


1.7 






72 


< K-9) 




K3.3 


1.9 


Sm 152 


26.63 


122 


1.4(-9) 


3.7(11) 


K0.7 


5.3 


Eu lsl 


47.77 


22 






L 12 


0.17 


Eu 153 


52.23 


84 








2.5 






97 


< K-9) 




K0.3 


3.3 






103 


3.4(-9) 


7.7(11) 


K 1.2 


3.7 


Gd 154 


2.15 


123 


1.2(-9) 


3.2(11) 


1.5 


5.3 


Gd lss 


14.7 


60 

87 

105 






K0.4 


1.2 
2.6 
3.8 


Gd 156 


20.47 


89 


2 (-9) 


3.9(11) 


K 1.0 


2.7 


Gd 160 


21.9 


75 








1.9 


Tb 159 


100 


58 


3.5(-ll) 


4.5(9) 


K6 


1.1 






137 


5.4(-ll) 


1.6(10) 




6.3 


Dy 160 


2.294 


87 


1.8(-9) 


3.4(11) 


K 1.5 


2.6 


Dy 161 


18.88 


25.7 


2.8(-8) 


1.6(12) 




0.22 






74.5 


3 (-9) 


4.9(11) 




1.8 


Dy 162 


25.53 


81 


3.2(-9) 


5.7(11) 




2.2 


Dy 163 


24.97 


75 








1.8 


Dy 164 


28.18 


73 


3.5(-9) 


5.6(11) 


K 2.7 


1.7 


Ho 165 


100 


95 


3.3(-ll) 


6.9(9) 


K 1.77 


2.9 


Er 164 


1.56 


91 


1.4(-9) 


2.8(11) 


K 1.9 


2.7 


Er 166 


33.4 


80 


1.8(-9) 


3.2(11) 


K 1.7 


2.1 


Er 168 


27.07 


79.8 


1.84(-9) 


3.2(11) 


K 2.1 


2.0 


Tm 169 


100 


8.4 


4 (-9) 


7.4(10) 




0.022 






118 


5 (-U) 


1.3(10) 


K 0.7 


4.4 


Yb 170 


3.03 


84.2 


1.57(-9) 


2.9(11) 


K 1.6 


2.2 



21.0 
1.0 
0.97 
0.67 
0.85 
1.7 

10.0 
0.71 
1.7 
3.1 
3.2 

30.0 
0.70 
1.1 
0.56 
0.63 
0.82 
4.8 
5.2 
4.6 
2.0 
0.70 
3.2 

10.0 
2.3 
3.3 
7.7 

22.0 
1.5 
2.6 
6.4 

37.0 
2.9 

19 
5.8 
6.2 
1.2 
5.1 
7.1 
6.2 
700 
3.1 
6.6 



(continued) 



from D. Strominger, J. M. Hollander, and G. T. Seaborg, Revs. 
Modern Physics, 30, 585 (1958), and C. L. McGinnis (ed.), ''Nuclear 
Data Sheets," National Academy of Sciences-National Research 
Council, Washington, D.C., 1958- . 



APPARATUS AND PROBLEMS 39 

Table 3-1 (continued) 



Stable 




Ey 


TV 2 , 












nuclide 


a, % 


kev 


sec 


Q 


a e/y 


R, 10~ 2 ev 


o\, 10~ 19 cm 2 


Effect 


Yb m 


14.31 


66.7 


< 5(-7) 






1.4 


11.0 




Yb 172 


21.82 


78.7 








1.9 


20.0 




Yb X73 


16.13 


78.7 








1.9 


5.2 




yb 174 


31.84 


76.5 








1.8 


21.0 




Lu 175 


94.4 


113.8 


8 (-11) 


2.0(10) 


K 1.6 


4.0 


0.90 




Hf 176 


5.21 


88.3 


1.35(-9) 


2.6(11) 


K1.32 


2.4 


6.7 




Hf 177 


18.5 


113 


4.2(-10) 


1.0(11) 


K 0.75 


3.9 


1.4 


X 


Hf 176 


27.1 


93.1 


1 (-9) 


2.0(11) 




2.6 


14.0 




Hf 180 


35.22 


93 


1.4(-9) 


2.9(11) 


KL4.0 


2.6 


2.8 




Ta 181 


100 


6.25 


6.8(-6) 


9.4(13) 


44 


0.012 


17 








136.1 


5.7(-ll) 


1.7(10) 


K 1.5 


5.5 


0.66 


X 


W "o 


0.135 102 






5 


3.1 


2.0 




W»2 


26.4 


100 


1.3(-9) 


2.9(11) 


4.5 


2.9 


2.2 


X 


W 1 " 


14.4 


46.5 
99.1 






9 
3.5 


0.63 
2.9 


2.3 
1.7 




W »4 


30.6 


111 


1.3(-9) 


3.2(11) 




3.6 


9.9 




W 186 


28.4 


123 


1.0(-9) 


2.7(11) 


K.0.45 


4.4 


5.6 




Re 185 


37.07 


125 






K 2.4 


4.5 


0.61 




Re 187 


62.93 


134 


2 (-9) 


5.9(11) 


K2.1 


5.2 


0.58 


X 


Os 186 


1.59 


137 


5.1(-10) 


1.5(11) 


K0.45 


5.4 


4.5 




Os 188 


13.3 


155 


6.2(-10) 


2.1(11) 


K 0.40 


6.8 


3.6 




Os 190 


26.4 


187 


3.5(-10) 


1.4(11) 


K0.2 


9.9 


2.9 




Os 192 


41.0 


206 


2.8(-10) 


1.3(11) 


K0.16 


12.0 


2.5 




Ir m 


38.5 


82.6 


3.9(-9) 


7.1(11) 




1.9 


1.8 








129 


1.4(-10) 


4.0(10) 


K2.9 


4.7 


0.56 


X 


Ir 193 


61.5 


73 


5.7(-9) 


9.1(11) 




1.5 


2.3 


X 






139 


2.6(-10) 


7.9(10) 


K2.2 


5.4 


0.59 




Pt 19S 


33.8 


99 






9.0 


2.7 


0.50 








129 


5.8(-10) 


1.6(11) 




4.6 


4.4 




Au 197 


100 


77 


1.9(-9) 


3.2(11) 


2.5 


1.6 


0.59 


X 


Hg 199 


16.84 


158 


2.4(-9) 


8.3(11) 


K0.2 


6.7 


2.4 




Hg 201 


13.22 


32.1 
167.6 








0.27 
7.5 


24.0 
0.44 





3-3 SOURCES AND ABSORBERS 

No time-honored and well-proved recipe exists according to which 
one can prepare sources and absorbers that yield Mossbauer lines as 
strong as possible and as sharp as the natural line width. A proce- 
dure leading to the best results has to be found for each isotope, 
mainly by trial and error. 36 In some cases, particularly in nuclides 
with extremely narrow lines, such as Ta 181 and Ge 73 , no resonance 
has been found as yet. All these difficulties are due to solid-state 
effects and technical problems, and one must learn more about these 



36 One valuable suggestion was put forward by S.S. Hanna at the 
second Mossbauer conference: "Find yourself a good chemist." 



40 THE MOSSBAUER EFFECT 

before he can predict with certainty the outcome of any given experi- 
ment. Because of these difficulties, there is little sense in writing 
down general guidelines; indeed, they would probably mislead ex- 
perimenters. Indications on source and absorber preparations can 
be found in nearly all the experimental papers reprinted in the back 
and listed in the bibliography. 37 

3-4 APPARATUS 

Two basically different ways of obtaining the velocity spectrum 
are in use. In one, the source (or the absorber) moves with a con- 
stant velocity for a pre-set time and the counts during this period 
are recorded. The velocity is then changed to a new value and the 
procedure is repeated until the entire spectrum is measured. 
Mossbauer's original work was performed in that way (Reprints 
Mossbauer 1958, 1959). 

In the second method, the source sweeps periodically through a 
range of velocities and the counts in predetermined ranges of vel- 
ocity are stored in different channels of a multichannel analyzer. 
These two methods gather information at the same rate, but each 
one has advantages for certain types of investigations. With con- 
stant velocity drives, small parts of the spectrum can be investi- 
gated with high accuracy. With the velocity sweep device, the 
entire spectrum is obtained simultaneously and a first impression 
can be obtained quickly. In either case, data from positive and neg- 
ative velocities of equal magnitude should not be lumped together 
without ascertaining that the velocity spectrum is symmetric with 
respect to velocity zero. 

One remark is in order about vibration. For isotopes like Au 197 , 
where Q = E /r ~ 5x 10 11 , vibrations are not very serious and any 
carefully built equipment will work. For nuclides like Fe 57 , where 
Q « 5 x 10 12 , the vibration problem is much more serious. As a 
rule of thumb, the following "scientific" observation can be useful: 
If one touches the equipment lightly with the fingertips and notices 
traces of vibration, the experiment will fail! Bubbling of liquid ni- 
trogen in dewars, for instance, will widen the lines considerably. 
For nuclides like Zn 67 , with Q » 10 15 , every trace of vibration must 
be carefully eliminated. 



37 The nuclide Fe 57 is probably the one that is of most interest 
to groups starting research and to instructors designing experi- 
ments for students. A very detailed description of the source 
preparation is contained in Margulies, 1962. Complete Co 57 sources 
and Fe 57 absorbers can be purchased from Nuclear Science and 
Engineering Co., Pittsburgh, Pa., and the U.S. Nuclear Corp., 
Burbank, California. 



APPARATUS AND PROBLEMS 



41 



3-41 Constant- Velocity Drives 

During the past three years, a number of constant- velocity drives 
have been constructed. None of these is so superior to all the others 
that it has replaced them, and physicists are still waiting for a sim- 
ple, accurate, easily controllable, and vibration-free drive. 

Three typical mechanical drives are shown in Fig. 3-2. The ro- 
tating disk used, for example, by DePasquali et al. 1960, Shirley et al. 
1961, and shown in Fig. 3-2a is probably the simplest system. The 






Fig. 3-2 Constant- velocity drives: (a) rotating disk; (b) three- step 
cam; (c) inclined plane. (From Margulies 1962.) 



42 THE MOSSBAUER EFFECT 

gamma ray passing through the disk sees in its direction of motion a 
velocity component raj sin 6, where r is the distance from the axis 
of rotation to the point where the gamma ray passes through the ab- 
sorber and 6 is the angle between the gamma ray and the axis of ro- 
tation. The slit shown in the figure can be designed in such a way 
that an increase in velocity due to an increase in r is compensated 
for by a decrease in sin 6. The velocity is changed by changing the 
angular velocity co. The main disadvantage of this system is the 
large area of absorber needed, (if the source is mounted on a wheel, 
the activity is used very inefficiently.) 

A cam, as shown in Fig. 3-2b, is quite reliable provided the drive 
is vibration-free. Usually, one uses photocells to switch the counting 
system off shortly before the follower on the cam reaches a maxi- 
mum or minimum and starts it again after passing these points. 
Counts "up" and "down" are stored in different scalers, and two 
more scalers count the corresponding times. 

The inclined plane (Argonne group), shown in Fig. 3- 2c, pro- 
vides a uniform motion with a very fine speed control. In this 
device a reversible synchronous motor is employed to drive a 
carriage by means of an accurately machined lead screw. Mounted 
rigidly to the moving carriage is a plane whose angle of inclina- 
tion can be continuously varied between and 45°. A shoe piv- 
oted to the foot of an extension rod slides smoothly along the in- 
clined plane, being held quite firmly to the plane by a thin layer 
of oil. As the carriage is driven back and forth, the extension 
rod moves to- and- fro, carrying the absorber with it. In addition 
to the fine speed control afforded by varying the incline angle, 
the lead- screw speed can be changed by a system of gears and 
pulleys not shown. 

These three systems are only examples; more possibilities for 
moving an object with constant velocity exists. An improved version 
of Fig. 3- 2b uses a heart-shaped cam and two followers. The car- 
riage is guided up and down and never coasts freely. Hydraulic de- 
vices can either be actuated by a hydraulic master connected to a 
rack- and- pinion drive (Reprint Pound and Rebka 1960) or by a sys- 
tem consisting of an oil pump and valves to control pressure and 
direction of the oil flow. Surplus servo cylinders from automatic 
pilot servo units, such as Electric Autolite Mark IV, make excellent 
and inexpensive hydraulic systems. 38 

Piezoelectric crystals and loudspeakers, when driven so that they 
execute a saw tooth or a triangular motion, are also versatile veloc- 
ity drives. Extreme care must be taken, however, to prevent res- 
onances in the system from distorting the wave shape. The best 



38 Such units are available, for instance, from Herbach and 
Rademan, Inc., Philadelphia, Pa. 



APPARATUS AND PROBLEMS 



43 



method consists in using a feedback mechanism involving a velocity- 
measuring device. 39 

3-42 Velocity Sweep Devices 

Velocity sweep systems, using a multichannel analyzer, permit 
one to investigate the entire velocity spectrum simultaneously (Re- 
print Ruby et al. 1960). The basic aspects of such an apparatus are 
shown in Fig. 3-3. 




liliih, 

modulated 
spectrum 




velocity spectrum 



Fig. 3-3 Velocity sweep system. The counts from the counter C 
are modulated in the modulator M by the velocity spec- 
trum v(t) and are then fed into the multichannel analyzer 
MA. The source S is mounted on the driver D; V is a 
velocity pick up and A is the resonant absorber. 



The source is moved in such a way that all desired velocities 
from 



max 



to +v are covered during one cycle. Attached to 
max 



the source is a velocity pickup which measures the instantaneous 
velocity v(t). This signal is used to route the energy- selected 
pulses from the counter into different channels of a multichannel 
analyzer. One way of doing it, not shown in Fig. 3-3, is to feed the 
signal v(t) directly into the address logic of a multichannel ana- 
lyzer. In the solution sketched in Fig. 3-3 the amplitude of the 
pulses from the counter is modulated by the signal v(t). An un- 
modified multichannel analyzer then sorts the pulses according to 
their velocity v(t). A third possibility is to use a multichannel 



The linearity of moving- coil loudspeakers has been investigated 
and reported on by J. Baumgardner, Argonne National Laboratory 
Report ANL-6169. 



44 THE MOSSBAUER EFFECT 

analyzer as a set of scalers and route the counts to the different 
channels by means of pulses. 

The simplest way of moving the source is to feed a sine wave into 
the driver. The velocity v(t) then varies sinusoidally with time. This 
choice of v(t) is easy to realize experimentally and one encounters 
few or no resonances in the driver. However, the source spends un- 
equal times at different velocities and the velocity spectrum must be 
correspondingly corrected. This problem can be avoided by selecting 
a linear velocity drive, as shown in Fig. 3-3. The motion x(t) of the 
source then must be a double parabola. Such motion must be care- 
fully controlled by feedback in order to prevent distortions. 

Velocity pickups can be constructed in different ways. The sim- 
plest one is to employ a loud speaker with two voice coils, use one 
for driving the speaker cone and the other to measure v(t). A simi- 
lar solution consists in coupling two speakers either by a mechanical 
link or through the air. A pickup coil can also be rigidly attached to 
the source and placed in a uniform magnetic field. A condenser or a 
commercially available velocity-measuring device 40 can also be used. 
All these velocity pickups can be used to route the pulses and to 
serve as feedback devices in controlling the motion of the driver. 

3-43 Special Velocity Drives 

In addition to the more conventional systems described in Sees. 
3-41 and 3-42, other possibilities exist. One is the "temperature 
drive/' If source and absorber have different temperatures, the 
emission and absorption lines will be shifted with respect to each 
other (cf. Sec. 5-4). This shift can be used to trace out a resonance 
line. Another possibility is provided by phase modulation, which is 
discussed in Sec. 5-5 (see also Reprint Grodzins and Phillips 1961). 
The relativistic drive offers a third possibility. The source is at the 
center of a rotor and the absorber on the rim. The transverse Dop- 
pler effect then provides a small and very accurately controllable 
shift (cf. Sec. 5-3). These three methods allow the investigation of 
very small shifts and are hence extremely well suited for the study 
of very narrow lines. 

3-5 CORRECTIONS 

Corrections and checks are necessary in order to ensure that the 
results of a Mossbauer experiment are correct. 

3-51 Background 

The background B due to other gamma rays and to X rays must 
be subtracted from the measured counting rates I exp in order to get 

40 E.g., LVsyn Transducers, Sanborn Co., Waltham, Mass. 



APPARATUS AND PROBLEMS 45 



the true counting rates l(v). Equation (59) then reads 

I H - I (v) 

*(v>= eX t P H ! X B P (60) 

i exp v ' D 

Mossbauer experiments are usually performed with gamma rays of 
energies below 100 kev, and X rays hence are often very disturbing. 
Very careful investigation of the pulse-height spectrum is necessary; 
additional information about the background can also be gained from 
absorption experiments. 

3-52 Source and Absorber Thickness 

The corrections for finite source and absorber thickness can be 
quite complicated, particularly if the lines do not possess a Lorent- 
zian shape and if they are split. Only a few remarks are given 
here, in order to outline some of the difficulties. 

The broadening of an absorption line in the Mossbauer effect due 
to finite absorber thickness has first been treated by W. M. Visscher 
(unpublished notes). He found that if emission and absorption lines 
have Lorentzian shapes of width T, the overlap curve will also be a 
Lorentzian but will show an apparent width r given by the relations 



T app /r = 2.00 + 0.27T ^ T < 5 

r app /r = 2.02 + 0.29T - 0.005T 2 4 & T & 10 



(61) 



Here T is the effective absorber thickness, given by 

T = f naa t (62) 

where f is the fraction of gamma rays absorbed without energy loss, 
n is the number of atoms per cubic centimeter, a the fractional 
abundance of the resonantly absorbing atoms, a the absorption cross 
section at resonance [i.e., Eq. (10) at resonance], and t the absorber 
thickness. Visscher' s relations are useful for quick estimates. 
Margulies and Ehrman (Reprint) have extended Visscher' s calcula- 
tions and their paper gives graphs for various cases of interest. 

If the shape of the emission and the absorption lines is given by 
a Gaussian distribution, the line broadening is much less pronounced 
than for Lorentzian shapes. A comparison of the line broadening for 
the two shapes is given in Fig. 3^4. It is clear from Fig. 3-4 that 
one must be extremely careful in extrapolations to zero absorber 
thickness. 

So far the discussion has been restricted to unsplit emission and 
absorption lines. In the case of hyper fine splitting, the analysis of 



46 



THE MOSSBAUER EFFECT 




- T = f'nacr o t 



Fig. 3-4 Ratio of apparent full width at half height to the 
widths of the emission and absorption lines as a 
functions of the effective absorber thickness T. 
The source is assumed to be thin (nonresonant). 
(Margulies 1962.) 

the transmission becomes more complicated but it can still be ob- 
tained from a straightforward generalization of the results given in 
the Reprint by Margulies and Ehrman (M-E) (see Margulies 1962). 
Some of the essential features can be summarized as follows: If only 
the emission line is split, and if Wi is the relative intensity of the 
i-th emission line, then equation 16 of M-E is still valid, provided f 
is replaced by fW^. If only the absorption line is split, the effective 
thickness of the resonance absorber corresponding to the j-th ab- 
sorption line is not T but W^T, where Wj is the relative intensity of 
the j-th absorption line. The absorber, for any given line, appears 
thinner than it really is. This fact must be kept in mind when deter- 
mining the line broadening from (61) or from Fig. 3-4. 



3-53 Velocity Resolution 

In any apparatus, the gamma rays will traverse the absorber at 
various angles, not just normal. This spread in angle introduces a 
spread in velocity which must be taken into account for accurate line- 
width determinations. 



APPARATUS AND PROBLEMS 47 

3-54 Reemission 

In most transmission experiments, one assumes that a resonant 
gamma ray which has been absorbed will not be registered in the 
counter. In general, this is not true and the amount of reradiation, 
which depends on the conversion coefficient and the solid angle of the 
counter, must be calculated and taken into account (Obenshain and 
Wegener 1961). 

3-6 USEFUL INFORMATION 

Some data useful in preparing and evaluating Mossbauer experi- 
ments are collected in this section. 

A nucleus of mass number A decays from an excited state with 
spin Ib, mean life r, or half- life T l/2 , to a stable ground state with 
spin Ia, and emits a gamma ray of energy E with a total conversion 
coefficient a. The recoil energy R of the free nucleus, the maximum 
resonance absorption cross section o' and the natural line width r 
are then given by 

R(ev) « 5.37 x 10" 4 E(kev)/A (63) 

2.45 x 10 9 21b + 1 1 



(64) 



E 2 (kev) 2I A + 1 1 + a 
r(ev) = 6.58 x 10" 16 /T(sec) 

= 4.55x 10- 16 /T 1/2 (sec) (65) 

Assume now that the nucleus is embedded in a solid and that the 
solid can be described in sufficient approximation by the Debye model, 
with a Debye temperature O. The fraction f of gamma rays emitted 
without energy loss to the lattice is then given by (52) and (53). These 
equations have been evaluated by Mossbauer and Wiedemann (1960) 
and by Cotton (1960). The curves presented in Fig. 3-5 were made by 
A. H. Muir, Jr. 41 

To determine f from Fig. 3-5 one calculates the ratios 0/T 
and R/0. T is the temperature of the solid and R is given (in ev) by 
Eq. (63). The curves in Fig. 3-5 are labeled by R/6 in units of 
ev/°K. 

The selection of an appropriate Debye temperature is not easy. 
First, published Debye temperatures (e.g.,Refs. 42 to 44)for any given 
substance vary widely, depending on the method of determination. 
Second, even if one finds a unique value in the literature, it is not cer- 
tain that it will fit the results from Mossbauer experiments. In Table 



41 A. H. Muir, Jr., Atomics International Report AI-6699 (1961) 



48 

f * 1 
0.8 
0.6 
0.4 

0.2 

0.1 
0.08 
0.06 
0.04 

0.02 
0.01 



THE MOSSBAUER EFFECT 





































P 


p*- 
















/ 1 


/ * 


/ 




<0 


/ <0 












/ / 


if 




' -•/ 


«v 


&' 




i 


/ 








V 




i 


7 




















i 


/ 
























/ 
















-* 


/ 


/ 




f 

1 1 


/ 




J 


i . 








^ 


/ 


/ 


i 


1 


/ 




/ 


/ 










/ 


/ 




/ 


/ 




/ 


/ 










/ 


/ 




/ 


/ 




/ 


/ 


/ 








/ 


/ 


I 


/ 


/ 




/ 


/ 


/ 








/ 


/ 


1 


/ 


/ 


/ 


1 


/ 


/ 










f 






f 






/ 


/ 









"^ to 00 

O © © r-l 



CO 00 

d © 



CO oo o 



Fig. 3-5 Graph for determining the fraction f as a func- 
tion of 0/T for various values of R/0. The 
parameter labeling each curve is R/0 in units 
of ev/°K. (From A. H. Muir, Jr., Atomics In- 
ternational Report AI-6699.) 

3-2 the Debye temperature of iron, as found in three compilations, is 
compared with the latest value obtained by the Mossbauer effect. This 
table shows that one has to be very careful in using published data 
when designing an experiment. 






42 "American Institute of Physics Handbook/' McGraw-Hill, New 
York, 1957, pp. 4-47 to 4-49. 

43 F. Seitz, "The Modern Theory of Solids," McGraw-Hill, New 
York, 1940, p. 110. 

44 M. W. Holm, "Debye Characteristic Temperatures,' ' Phillips 
Petroleum Co. Report IDO-16399, Office of Technical Services, U.S. 
Department of Commerce, Washington 25, D. C. 

45 S. S. Hanna, in "Proceedings of the Second Mossbauer Conference,' 
A. Schoen and D. M. J. Compton (eds.), Wiley, New York, tentative pub- 
lication Spring 1962. 



APPARATUS AND PROBLEMS 49 

Table 3-2 
Comparison of Debye Temperature of Iron Found by 
Conventional Methods with That Deduced from the Mossbauer Effect 

Method 0,°K Ref. 



Conventional 


355 


42 




420 


43 




420-460 


44 


Mossbauer 


490 


45 




NUCLEAR PROPERTIES 



4-1 INTRODUCTION 

In many ways, investigations on the Mossbauer effect follow a path 
along which nuclear resonance experiments have been going. The dis- 
covery of the Mossbauer effect grew out of the study of a nuclear prop- 
erty, namely, the lifetime of the first excited state in Ir 191 . This goal 
was attained, but at the same time a powerful tool for solving other 
problems was created. Indeed, the main applications in the past two 
years have not been to nuclear physics, but to other fields, such as 
relativity and solid-state physics. Despite this development in an un- 
expected direction, applications to nuclear physics remain challenging. 
A few of these are sketched in Sees. 4-2 to 4-5. 

Many of the discussions in this and the following chapters refer to 
Fe 57 , probably the most used of the "Mossbauer nuclides." For this 
reason the decay scheme of Fe 57 is shown in Fig. 4-1. 

270-day Co 57 

electron capture 
0.137 Mev 




91% 



0.01437 Mev T = 1.4 x 10" 7 sec 
stable ground state of Fe 57 



Fig. 4-1 Decay scheme of Fe 57 . The 14.4-kev transition 
3/2 — 1/2 is the one used in many Mossbauer 
experiments. 



50 



NUCLEAR PROPERTIES 51 

4-2 LIFETIME AND CONVERSION COEFFICIENT 

In conventional nuclear resonance fluorescence experiments 10 " 12 
one can measure the scattering cross section and is able to deduce 
the gamma-ray width Ty by using (9) modified to take into account 
the Doppler broadening. This procedure is rather indirect, and the 
Mdssbauer effect permits a more straightforward approach, yielding 
both T and IV. The total line width r is found by tracing out the ab- 
sorption or the scattering line by using the Doppler effect, as sketched 
in Fig. 3-1. The ratio of the scattering and the absorption cross sec- 
tion, Eqs. (9) and (10), yields Ty/T . From the total line width one im- 
mediately gets the total lifetime r by (3) or (65). From the ratio 
Ty/T , one finds the total conversion coefficient a by (13). 

The measurement of the line width to determine the lifetime was 
first used by Mossbauer (Reprint 1958) in the case of Ir 191 . In a later 
investigation, Mossbauer and Wiedemann (1960) were able to find a 
value of t = (1.5 ± 0.2) X 10" 11 sec for the lifetime of the 134-kev ex- 
cited state in Re 187 . This lifetime is at present the shortest one that 
has been determined by the Mossbauer effect. 

Despite the simplicity of this method, some difficulties exist. For 
lifetimes of the order of 10" 10 sec or longer, extranuclear fields can 
widen the line, and one may find apparent lifetimes that are shorter 
than the real ones. The 14.4-kev transition in Fe 57 (Fig. 4-1) offers 
an excellent example for these difficulties. The lifetime of the 3/2 
state is known from electronic measurements to be r = 1.4 xl0" 7 sec. 
According to (65) the corresponding line width is 4.7 xlO -9 ev. All the 
early experiments, however, yielded line widths considerably larger 
than that. Only after the source and absorber preparation was care- 
fully investigated did experiments yield values T e xp within 10 per 
cent of the expected one. Hence very careful measurements of the 
line shape and a thorough understanding of the solid-state processes 
involved in the line widening are prerequisites for valid estimates of 
lifetimes longer than, say, 10" 10 sec. 

Difficulties may also arise for very short lifetimes. If the nuclear 
lifetime becomes comparable to, or shorter than, the inverse of the 
Debye frequency (~ 10" 13 sec), it may be impossible to distinguish re- 
coilless transitions from those in which the lattice takes up energy. 
Furthermore, heat spikes due to previous decays may well destroy 
the effect. Investigations in this range are very desirable and they 
will probably yield information not only about nuclear lifetimes, but 
also about solid-state processes connected with nuclear decays. 

4-3 NUCLEAR MOMENTS 

The Mossbauer effect permits the determination of nuclear mo- 
ments, i.e., spin, magnetic dipole moment, and electric quadrupole 
moment, in a rather direct way, provided certain conditions are met. 



52 THE MOSSBAUER EFFECT 

Consider as the simplest example a nuclide with spin in the ground 
state A and spin I B , magnetic moment /x B , and g factor g B in the 
excited state B. Assume further that emitting and absorbing nuclei 
are embedded in solids in such a way that the nuclei see no magnetic 
field or electric field gradient. The emission and absorption lines 
will be unsplit. If one now applies an external magnetic field H to 
source or absorber, the corresponding line will split into 2I B + 1 
equally spaced components, with a separation between components of 

A=g B M H (66) 

Here /i is the nuclear magneton, /j. = 5.05 x 10" 24 erg/gauss 

= 3.15 x 10" 12 ev/gauss. If the separation is larger than the width 2T 

of the overlap line (Fig. 3-1), i.e., if 

g B M H 2 2T = 2ti/r (67) 

the number of components can be counted and one has found the spin 
I B . From the splitting A, one determines the nuclear g factor g B if 
the external field is known. The nuclear g factor and the spin to- 
gether yield the magnetic moment, ji B = gBMo^B- As an example for 
condition (67), take g B = 1, H = 10 5 gauss. A separation is then possi- 
ble for r £ 6 x 10" 9 sec. 

If condition (67) is not fulfilled, the spin I B cannot be determined 
simply by counting the number of components. Nevertheless it is often 
possible to find an approximate value of the g factor by measuring the 
increase in line width as a function of the applied field H. 

The situation encountered usually is more complicated than out- 
lined above, since the ground-state spin will in general not be zero 
and very large internal fields are often present. The first problem, 
nonzero ground- state spin, results in the observation of more than 
2I B + 1 components. Usually, however, the ground- state spin and mo- 
ments can be measured with conventional techniques and their knowl- 
edge permits one to unravel the complex spectrum. Moreover, polar- 
ization experiments can help to reduce the number of components and 
to make their identification easier (see Sec. 5-9). 

The presence of very strong internal magnetic fields in ferromag- 
netic and paramagnetic substances introduces complications on the 
one hand, but, on the other, allows the determination of g factors that 
otherwise would be difficult to find. Indeed, the first observation of 
the Zeeman effect of a nuclear gamma ray was performed by using 
the internal magnetic field in Fe 57 (Pound and Rebka 1959a, DePasquali 
et al. 1960, Reprint Hanna et al. 1960). Two limiting cases must be 
distinguished when considering internal fields. One extreme occurs 
when the field can be considered to be a static external field; the num- 
ber and splitting of components observed in a given direction is then 



NUCLEAR PROPERTIES 53 

given by the nuclear moments and the direction and magnitude of the 
magnetic field. This case is approached, for instance, by Fe 57 in iron. 
The other extreme corresponds to free atoms where the hyperfine in- 
teraction is governed by the usual spectroscopic rules: If the atomic 
spin is denoted by J, the multiplicity of components depends on J also. 
For instance, if J <Ia> the ground state splits only into 2 J + 1 mag- 
netic sublevels. Obviously, a free atom cannot exhibit a Mossbauer 
effect. However, in the rare earth elements situations can arise 
where transitions without energy loss occur and where at the same 
time the unfilled inner shells can be treated like free atoms. 

To investigate the quadrupole coupling Q- VE, where Q is the nu- 
clear quadrupole moment and VE the electric field gradient at the 
nuclear site, one must place source or absorber nuclei in a surround- 
ing with a high electric field gradient VE and measure the resulting 
splitting. If one wants to find the quadrupole moment Q, one must 
calculate the field gradient. 

The first experimental investigation of quadrupole effects was done 
by Kistner and Sunyar, using Fe 57 . Fe 57 is well suited for this purpose, 
since the ground- state spin 1/2 prevents the ground state from having 
a quadrupole moment, and only the splitting due to the excited state 
with spin 3/2 must be taken into account. Kistner and Sunyar showed 
that the splitting of Fe 57 embedded in antiferromagnetic Fe 2 3 can be 
explained by a superposition of magnetic hyperfine structure and quad- 
rupole interaction (Reprint Kistner and Sunyar 1960). In the meantime, 
many examples have been found in which only quadrupole splitting ex- 
ists (see, for instance, Reprint DeBenedetti 1961). 

The number of publications reporting investigations of magnetic and 
quadrupole interactions in various systems with various nuclides is 
growing steadily, and the reader is referred to the bibliography. 

Incidentally, it is interesting to note that the measurements of mo- 
ments of excited states by the Mossbauer effect have already led to 
additional theoretical investigations. Attempts to derive the quadru- 
pole moment of the 3/2 state in Fe 57 have been made, 46 ' 47 and the 
magnetic moment of the 3/2 state has turned out to be a sensitive 
measure for the validity of nuclear model calculations. 48 (See also 
Abragam 1961, Gastebois 1961.) 

4-4 ISOMERIC SHIFTS (CHEMICAL SHIFTS) 49 

In atomic spectroscopy, lines from an isotopic mixture show a 
splitting which is not present in the spectrum of an isotopically pure 



46 R. Bersohn, Phys. Rev. Letters, 4, 609 (1960). 
47 G. Burns, Phys. Rev., 124, 524 (1961). 



48 R. D. Lawson and M. H. Macfar lane, Nuclear Phys., 24, 18 (1961). 
49 Both terms, isomeric and chemical shift, are used in the litera- 
ture to denote the same effect. 



54 THE MOSSBAUER EFFECT 

element. In heavy elements, this isotopic splitting is due to the fact 
that the addition of one or more neutrons changes the nuclear radius. 
This change, in turn, shifts the atomic energy levels. 50 ' 55 A change in 
the radius can occur even without a change in nucleon number when 
the nucleus goes from one state to another, for instance, when it de- 
cays from an isomeric state to the ground state. The corresponding 
shift in energy is called an isomeric shift and such a shift of atomic 
energy levels has recently been observed in Hg 197 . 56 

A change in nuclear radius which shifts the atomic energy levels 
will obviously also affect the nuclear levels by the same amount. An 
isomeric shift has indeed been observed in the Mossbauer effect and 
this shift may well turn out to yield more information about nuclear 
and solid-state physics than, for instance, the lifetime measurements 
described in Sec. 4-2. The first unambiguous observation of an iso- 
meric shift was made by Kistner and Sunyar (Reprint 1960), using 
Fe 57 . Subsequently, isomeric shifts were investigated with Fe 57 (e.g., 
Solomon 1960; Reprint DeBenedetti 1960, Wertheim 1961); with Sn 119 
(Boyle 1961); and with Au 197 (Shirley 1961). Systematic discussions of 
the isomeric shifts have been published by Walker et al. (Reprint 1961) 
and by Shirley (1961). 

In the present section the physical idea underlying the isomer shift 
will be outlined in simplest terms. As pointed out above, a change in 
the electrostatic interaction between the nucleus and its electron shell 
is responsible for the equal energy shifts of the atomic and the nuclear 
levels. The observation of these two effects is entirely different, how- 
ever. Consider first the atomic isotope and isomer shift. Isotopes, 
and isomers have the same electron shell but different nuclear radii. 
The same atomic transition hence can have a different energy in two 
atoms which contain nuclei in different states, provided the two atomic 
states involved in the transition are affected differently by the change 
in nuclear radius. Otherwise both levels are shifted by equal amounts 
and no net change in the transition energy results. In the nuclear iso- 
mer shift, one compares the same nuclear transition in two atomic 
systems which have different electronic wave functions at the nucleus. 



50 H. Kopfermann, "Kernmomente," Akademische Verlagsgesell- 
schaft, Frankfurt, 1956; "Nuclear Moments," Academic Press, New 
York, 1958. 

51 K. W. Ford and D. L. Hill, Ann. Rev. Nuclear Sci., 5, 25 (1955). 

52 J. E. Mack and H. Arroe, Ann. Rev. Nuclear Sci., 6, 117 (1956). 

53 D. L. Hill, Matter and Charge Distribution within Atomic Nuclei, 
"Encyclopedia of Physics," Springer, Berlin, 1957, Vol. 39, p. 178. 

54 G. Breit, Revs. Modern Phys., 30, 507 (1958). 

55 L. R. B. Elton, "Nuclear Sizes," Oxford University Press, New 
York, 1961. 

56 A. C. Melissinos and S. P. Davis, Phys. Rev., 115, 130 (1959). 



NUCLEAR PROPERTIES 



55 



A shift can only be seen if the two nuclear states involved have differ- 
ent radii; otherwise both levels involved in the transition change by 
the same amount, and the gamma-ray energy is not affected. The re- 
quirements for the observation of a nuclear isomeric shift hence are: 

1. The two nuclear states involved must have different charge dis- 
tributions. 

2. There must be electronic wave functions (usually from s elec- 
trons) which overlap appreciably with the nuclear wave functions. 

3. These wave functions must be sensitive to external (chemical) 
changes. 

All three requirements are justified by the following simple calcu- 
lation. 

In order to calculate the shift of the nuclear energy levels, consider 
first a point nucleus, of charge Ze, with two levels A and B, sepa- 
rated by an energy E . The electrostatic potential created by this 
point nucleus is shown as a dotted line in Fig. 4-2. Now consider an- 
other nucleus, of equal charge, but with radius Ra in the ground state 
A and a radius Rg in the excited state B, as shown in Fig. 4-2. Be- 
cause of the diminished interaction with the electrons, the two levels 
A and B will lie higher for this nucleus than they did for the point 



R A R B 



► r 



u 



^"constant 
potential 
nucleus 



V(r) 
Fig. 4-2 



Ze/r 



-_IE E B 



Ej 



1AE A 



Electrostatic potential for a point nucleus and 
for a constant potential nucleus . The radius Ra 
of the ground state is assumed to be smaller 
than the radius Rg for the excited state. The 
level shifts from point nucleus to finite- size 
nucleus are shown on the right. 



nucleus (Fig. 4-2). To find for instance the shift AE^ one determines 
the contribution to the total energy of the system which comes from 
the electrostatic interaction. The potential from Ra to infinity is the 
same for the finite as for the point nucleus. Hence only the volume 
from to Ra needs to be taken into account when determining the en- 
ergy shift. If one assumes that the wave function ^(r) of the relevant 
electrons is essentially constant over the distances involved and can 
be replaced by ^(0), the contribution to the interaction energy from 
this volume is given by 



56 THE MOSSBAUER EFFECT 

E(Ra) = - / y ( r ) e l^(°)| 2 47ri * 2 dr (68) 

o 

where V(r) is the electrostatic potential created by the nucleus. For 
a point nucleus, V(r) = Ze/r, and one finds 

E P (RA) = -2;rZe 2 R 2 A |*(0)| 2 (69) 

The energy E(R^) for a finite nucleus depends on the charge distri- 
bution. Easiest to calculate is the surface charge model (top-slice 
model), where the electrostatic potential is constant from the center 
to the nuclear surface and joins the outside potential at r = R. This 
potential is shown in Fig. 4-2, and its contribution is given by 

E g (R A ) = - (4/3)77 Ze 2 R 2 A |*(0) | 2 (70) 

The difference between the expressions (70) and (69) yields the shift 

AE A = E g (R A ) - E p (R A ) = (2/3)77 Ze 2 R 2 A |^(0) | 2 (71) 

The assumption of a surface charge is unrealistic, but the calculation 
can easily be performed with a nuclear charge density p(r). The re- 
sult has exactly the same form as Eq. (71), but the nuclear radius R^ 
is replaced by the root-mean-square radius 

< R A> = /p( r ) r2 dv = 477 /p(r)r 4 dr (72) 

The transition energy between two levels B and A becomes 

EJ = E + AE B - AE A = E +(2/3) 77Ze 2 |*(0)| 2 [<Ra>- < r b>] 

(73) 

Ordinarily, it is impossible to observe such a small energy change. 
However, if one performs a Mossbauer experiment and uses source 
and absorber with different chemical environments such that the wave 
functions at the nuclei in the emitter e and the absorber a are differ- 
ent, one finds for the difference in gamma-ray energy from (73) 

6=E a -E e = (2/3)7rZe 2 [<R 2 B >-<R 2 A >]{|^(0) a | 2 -|^(0) e | 2 } (74) 

Equation (74) justifies the three requirements listed above, and it 
gives the correct order of magnitude for the isomeric shift. However, 
for many applications a more accurate formula is desired. Relevant 
derivations and equations can be found in refs. 50 to 55 and 57, and in 



57 



A. R. Bodmer, Nuclear Phys., 21, 347 (1960), 



NUCLEAR PROPERTIES 57 

the publication by Shirley (1961). Applications of (74) are contained in 
the Reprints by DeBenedetti et al. and Walker et al. 

Equation (74) also shows that information on two different fields 
can be obtained from isomeric-shift experiments. On the one hand, 
differences in the nuclear radius between ground state and excited 
state can be measured. On the other hand, changes in the wave func- 
tion of s electrons can be studied in various substances. Both ave- 
nues offer exciting possibilities for further investigations. 

4-5 PARITY EXPERIMENTS 

The observation of the intensity and of the polarization of individual 
Zeeman components of nuclear gamma rays (Sees. 4-2 and 5-9) per- 
mits the investigation of parity conservation in strong and weak inter- 
actions. Grodzins and Genovese (1961) have tested the parity conser- 
vation in strong interactions in the 14.4-kev transition in Fe 57 by com- 
paring the transition rates for the Am = ± 1 components in the direc- 
tion of the nuclear polarization with those opposite to it. They found 
the relative strength of a parity admixed wave function to be less 
than lO -5 . 

Morita (1961) has suggested experiments testing parity conserva- 
tion and time-reversal invariance in weak interactions. 




GENERAL PHYSICS 



5-1 SURVEY 



Those applications of the Mossbauer effect that are not nuclear 
or solid-state physics are reviewed in this chapter. When going 
through the various sections, one realizes that such a division is very 
arbitrary and that it should only be understood as a scheme for ar- 
ranging things. Actually, one of the beautiful aspects of the Moss- 
bauer effect is that it shows the unity of physics and connects seem- 
ingly unrelated fields. 

Comparing the applications in Sees. 5-2 to 5-5 with those of Chaps. 
4 and 6, one notices that they belong to a different class. Indeed, 
rather than divide the various topics as is done in this book, one can 
equally well distinguish two classes: (1) experiments depending on the 
line width T, and (2) experiments depending on Q = E/r, the ratio of 
the transition energy to the line width. 

It is really for class 2 that the Mossbauer effect shows its superi- 
ority as a research tool; it is surprising that more experiments have 
not been done which take advantage of the extremely high Q values 
available with nuclides such as Fe 57 and Zn 67 . 

5-2 GRAVITATIONAL RED SHIFT (APPARENT WEIGHT OF 
PHOTONS) 

Einstein's theory of relativity has always held a particular fasci- 
nation for the scientist and the layman alike, and it has stirred imag- 
ination as no other theory has done in the present century. It is there- 
fore not surprising that the use of the Mossbauer effect to study prob- 
lems connected with relativity has received an enormous amount of 
attention and publicity. Soon after the discovery of the Mossbauer ef- 
fect, it was realized by many physicists that this new tool offered pos- 



58 



GENERAL PHYSICS 59 



sibilities for investigating problems in relativity. A few groups 
started to explore the feasibility of such experiments. No group has 
done more admirable work, however, than Pound and his co-workers 
at Harvard. They have succeeded in unambiguously demonstrating the 
red shift of photons in the laboratory. 

The red shift of spectral lines in a gravitational field was pre- 
dicted by Einstein 58 ' 59 and a classic description of the relevant facts 
can be found in Pauli's article 60 (see also Moller 1960). Here a short 
outline of the essential arguments will be given. Consider a source of 
radiation at rest in a system Kq, which moves with a velocity v= /3c 
with respect to the laboratory system K^. The radiation source can 
be used as a clock. If this clock measures in its system Kq a time t 
(proper time) between two events, then an observer in K^^ will find 
a longer time tj a ^ between these same events (time dilation): 

t lab = V(l - 2 )V2 (75) 

If the source in Kq emits a radiation of frequency oj , then the fre- 
quency measured in K^ is given by 61 

± (1 - /3 2 ) 1/2 . 1 + g cos a (na , 

^ab " -o j . - p cos fllab " -o (1 . ^ ™ 

In these expressions for the Doppler shift, a is the angle between 
the direction of emission of the photon oj and the velocity v measured 
in Kq, and cq a b denotes the same angle measured in K^. 

For small velocities, v«c, (75) and (76) can be approximated by 

tlab ■* U 1 + (!/2)/3 2 ] (77) 

and 

w lab * ^ot 1 + cos a lab - (l/2)/3 2 ] (78) 

Now consider a source which is fixed at a radius R to a disk 
which rotates with an angular frequency £2. An observer at the cen- 
ter of the disk will receive only photons that have been emitted radi- 



58 A. Einstein, Ann. Physik, 35, 898 (1911); reprinted in "The 
Principle of Relativity," Dover, New York, 1923. 

59 A. Einstein, "Uber die spezielle und allgemeine Relativitats- 
theorie," Vieweg, Braunschweig, 1956, 17th ed., p. 82; "Relativity, the 
Special and the General Theory," Methuen, London, 1954. 

60 W. Pauli, "Theory of Relativity," Pergamon, London, 1958, 
pp. 19, 151-154. 

61 E. L. Hill, Optics and Relativity Theory, in E. U. Condon and H. Odi- 
shaw (eds.), "Handbook of Physics," McGraw-Hill, New York, 1958. 



60 THE MOSSBAUER EFFECT 

ally, i.e., for which cq a b = 90°. According to Eq. (78), the observer 
then measures a frequency 

^lab - co [l - (l/2)(Rft/c) 2 ] (79) 

This relation also follows from the time dilation (77), since an in- 
crease in the time interval between two signals entails a correspond- 
ing decrease in energy. 

The arguments leading to Eq. (79) are based on the Doppler effect; 
the observer determines the change in frequency due to the trans- 
verse motion of the source. Einstein has shown that the frequency 
shift can be considered from a different point of view— one which leads 
to an extremely important generalization. Instead of postulating a 
transverse Doppler effect, the observer can assume that his disk is 
at rest and that the source instead is in a different gravitational po- 
tential. The centrifugal force acting on the source then is interpreted 
as a gravitational force. The potential at the radius R is equal to the 
negative of the work required to move a unit mass from R to the 
center: 

<J>=-(1/2)R 2 S2 2 (80) 

Equation (79) can now be written as 

^lab ~ <o (l + $/c 2 ) (81) 

Einstein postulated Eq. (81) to hold not only for the rotating disk but 
generally for all situations where observer and clock are situated in 
different gravitational potentials. Whether the potential is gravita- 
tional or due to uniform acceleration does not matter (principle of 
equivalence). 

The derivation of Eq. (81) shows that the gravitational red shift 
does not yield information about the equations of general relativity. It 
only tests the principle of equivalence and shows that photons in a 
gravitational field behave like particles with mass. Pound correspond- 
ingly terms the terrestrial-red-shift experiment "apparent weight of 
photons." A particle with mass m gains an energy m<£, if moved 
from a potential to a potential <£. According to Eq. (81), a photon 
of energy E = na; gains an amount "fico $/c 2 . The photon thus behaves 
like a particle with a "mass" 

m photon=Wc 2 = E/c 2 (82) 

As expected, the rest mass of photons is zero. One could, of course, 
start from (82) and arrive at (81). 

If a source of resonant gamma rays is situated in the earth's gravi- 



GENERAL PHYSICS 61 



tational field at a height H above a resonant absorber, then the poten- 
tial difference is given by <£>= gH, as long as H is small compared to 
the radius of the earth. A photon, emitted by this source, will "fall" 
in the gravitational field toward the absorber and actually undergo a 
"violet" rather than a red shift. The corresponding relative gain in 
energy is given in first approximation by (81) as 

AE/E = (wiab " w )/o>o = gH/c 2 (83) 

or 

AE/E = H(cm) x 1.09 x 10" 18 (84) 

For a difference in height between source and absorber of 10 m, one 
gets a relative shift of about 1 part in 10 15 . 

One aspect of the red-shift experiment deserves a few more 
words. The arguments given above assume that the frequency of 
photons depends on the gravitational potential but that the nuclear 
levels, measured by an observer at rest with respect to the nucleus 
and situated in the same gravitational potential, remain unchanged. 
This assumption is based on the weakness of the gravitational inter- 
action as compared to nuclear and electromagnetic forces. It is, 
however, possible to select a coordinate frame in which the fre- 
quency of the photons moving in a static gravitational field remains 
constant (Miller 1960, p. 46). Measured in this coordinate frame, 
the energy of the atomic or nuclear levels depends on the gravita- 
tional potential, and it is this dependence that is responsible for the 
red shift. These two descriptions are equivalent. 

The basic idea of a terrestrial-red- shift experiment is very sim- 
ple. A resonant source is placed at a height H above the resonant ab- 
sorber, and the relative energy shift is determined by measuring the 
velocity spectrum. However, the practical difficulties are enormous, 
particularly because even Fe 57 , the nuclide used at the present time, 
has a Q value of "only" 3 x 10 12 . For a height H = 10 m, a shift of 
about one part in 10 3 of the line width must be determined accurately. 
In addition, many disturbing effects exist and must be eliminated or 
corrected for with extreme care. 

The first completely satisfactory measurement was performed by 
Pound and Rebka (Reprint 1960), and the reader is referred to their 
excellent report and to Pound's review (1960, 1961) for an understand- 
ing and appreciation of this difficult experiment. 

Two additional remarks are in order. During the past few years, 
physicists have often speculated whether antiparticles will fall or rise 
in a gravitational field. Winterberg has pointed out recently 62 that the 



62 



F. Winterberg, Nuovo cimento, 19, 186 (1961), 



62 THE MOSSBAUER EFFECT 

positive result of the Pound-Rebka experiment makes it extremely 
likely that antiparticles and particles have the same sign of the gravi- 
tational interaction. Consider the following gedanken experiment. A 
photon of energy 2m c 2 - e falls in a gravitational field until it has 
gained an energy € . In the presence of a very heavy nucleus, it then 
creates a particle-antiparticle pair of total mass 2mc 2 . If the anti- 
particle is subject to "anti gravity/' the pair is weightless and can 
be brought to the starting point of the photon without energy loss, thus 
violating energy conservation. 

The second remark concerns a suggestion by Cocconi and Salpeter 
(1960), who proposed to use the Mossbauer effect to set an upper limit 
on a possible anisotropy of inertia. Corresponding experiments using 
the Mossbauer effect (Sherwin 1960) and conventional resonance tech- 
niques 63 ' 64 failed to observe an effect and hence apparently deter- 
mined a limit on the anisotropy of inertia. Dicke, however, points out 
that this null result does not cast doubts on the validity of Mach's 
principle, but shows that inertial anisotropy effects are universal and 
the same for all particles. 65 

5-3 ACCELERATED SYSTEMS 

The high Q value of isotopes like Fe 57 allows the observations of 
effects due to accelerated systems. The simplest such experiment is 
the measurement of the energy shift of a photon emitted from a sys- 
tem rotating with a frequency ft. If a resonant emitter is mounted on 
this system at a radius R e and a resonant absorber is situated at a 
radius Ra> one finds for the relative energy shift from Eqs. (79) or 
(81), in first approximation, 

(E a - E e )/E = (ft 2 /2c 2 )(R| - R|) (85) 

At first one would expect that source and absorber must lie on the 
same radius or that the absorber must be exactly in the center to 
avoid a linear Doppler effect. However, a short calculation shows 
that the first-order terms will cancel. Such a cancellation is to be 
expected, since the gravitational field equivalent to the acceleration 
on a rotating system depends only on the distance from the center 
[Eq. (80)] . 

The Harwell group (Reprint Hay 1960) first demonstrated experi- 
mentally that Eq. (85) is followed within the limits of error. This 
experiment, together with the Pound-Rebka experiment discussed in 



63 V. W. Hughes, H. G. Robinson, and V. Be ltrow- Lopez, Phys. 
Rev. Letters, 4, 342 (1960). 

64 R. W. P. Drever, Phil. Mag., 6, 683 (1961). 
65 R. H. Dicke, Phys. Rev. Letters, 7, 359 (1961). 



GENERAL PHYSICS 63 



Sec. 5-2, shows that the equivalence principle is correct to within 
about 5 per cent. 

Equation (85) indicates that no shift is to be expected if source and 
absorber move on the same orbit, even if their relative velocity is as 
high as 2RS2. This absence of a Doppler shift has been verified by 
Champeney and Moon (Reprint 1961). 

Bommel has performed an experiment in which he tested the fre- 
quency shift in a linearly accelerated system 66 by mounting source 
and absorber on piezoelectric crystals and driving the crystals in 
phase. 

5-4 SECOND-ORDER DOPPLER SHIFT 

In the framework of the classical theory of the Mossbauer effect, 
as sketched in Sec. 2-3, Eq. (78) can be interpreted as follows. The 
emitting nucleus moves about its equilibrium position with a velocity 
v(t). As discussed in remark 4 of Sec. 2-3, the characteristic time 
for the lattice vibrations is much shorter than the lifetime of the nu- 
clear state. The linear term (v/c) cos cq a b in Eq. (78) will hence 
average out, giving rise to the unshifted and sharp Mossbauer line. 
The quadratic term, -(l/2)(v/c) 2 , will remain and it will cause a shift 
in the energy of the emitted or absorbed gamma ray (second-order 
Doppler shift). This shift can be calculated, since (l/2)Mv 2 = E kin , 
where M is the mass and E kul the kinetic energy of the decaying 
atom. Hence one finds 

6E/E = -E kin /Mc 2 (86) 

In the classical limit, the kinetic energy E km is equal to (3/2)kT, and 
the relative energy shift becomes 

6E/E = - 3kT/2Mc 2 (87) 

For harmonic lattice forces, where the total energy U is twice the 
kinetic energy E k j n , one finds for the temperature dependence of the 
relative energy shift from Eq. (86), 

-—- (6E/E) = -C L /M'c 2 (88) 

Here Cl = 8U/3T is the specific heat of the lattice and M' is the 
gram atomic weight of the lattice substance. 

The question of whether the temperature motion that causes the 
shift (86) will also give rise to a broadening of the lines has been 



66 H. Bommel, in "Proceedings of the Second Mossbauer Confer- 
ence, " Paris, Sept. 13-16,1961, A. Schoen and D. M. J. Compton (eds.), 
Wiley, New York, tentative publication Spring 1962. 



64 THE MOSSBAUER EFFECT 

studied by Snyder and Wick (1960). They find that for a perfect crys- 
talline solid such a broadening does not occur. 

If the environment of source and absorber nuclei are the same, the 
term (86) will be identical for both and hence will not be observable. 
However, if the source and the absorber are at different temperatures 
or if source and absorber nuclei are in different surroundings, an ob- 
servable shift can result. Since such a shift depends on the ''chemical 
state" of source and absorber, it is sometimes difficult to separate it 
from the isomeric shift discussed in Sec. 4-4. 

Actually, the second-order Doppler shift was overlooked until early 
1960, when Pound and Rebka (Reprint 1960) realized that such an effect 
existed. They derived an expression for it and verified this expression 
[Eq. (88)] experimentally. At the same time and from a different point 
of view, the effect was predicted by Josephson (Reprint 1960). In an un- 
dergraduate examination at Oxford University, Josephson received the 
problem to calculate the change in frequency of an oscillator which 
suddenly changes its mass. He had read about the Mossbauer effect 
and realized that there was a connection. When the excited state de- 
cays by gamma-ray emission, the nucleus loses energy, and its mass 
is reduced by an amount 5M = -E/c 2 . Its thermal momentum p is 
unchanged, since the solid takes up all the recoil momentum. The re- 
duced mass, however, causes an increase in the kinetic energy of the 
atom: 

5E kin = (8E kin /9M) 6M = (-p 2 /2M 2 )(-E/c 2 ) = (l/2)E(v 2 /c 2 ) (89) 

which is compensated by a corresponding decrease in the energy of 
the emitted gamma ray. The shift Eq. (89) thus agrees in magnitude 
and sign with the classical result deduced from Eq. (78). Josephson 
wrote a short note on his derivation and sent it to Harwell, where it 
arrived among many crank letters. Marshall realized the importance 
of Josephson's calculation and tried to reach him by telephone, only 
to be told that undergraduates were not supposed to receive calls. 
Despite these difficulties, Josephson's note was published. 

The experiments confirming the second-order Doppler shift in Fe 57 
(Reprint Pound and Rebka 1960) and Sn 119 (Boyle 1960) also constitute 
a direct proof of the famous "clock paradox" or "twin paradox," as 
was pointed out by Sherwin (1960). In his original paper on special 
relativity, Einstein 67 predicted the following effect: At a time t = 0, 
two identical clocks are synchronized at a point A. One clock remains 
at A and measures the time t. The other clock travels away from A 
with a velocity v(t). When it eventually returns to A, it will indicate 
an elapsed time s which is shorter than the time t measured by the 
stationary clock in the ratio 

67 A. Einstein, Ann. Physik, 17, 891 (1905). Reprinted in "The 
Principle of Relativity," Dover, New York, 1923, p. 49. 



GENERAL PHYSICS 65 

to 
So/to = UAo) / [l-(v/c)T 2d t (90) 



Equation (90) is a generalization of Eq. (75); in one form or another 
it is an integral part of science fiction literature. The paradox lies in 
the fact that a straightforward application of ' 'relativity/ ' without a 
careful definition of its meaning, leads to the conclusion that one 
should never be able to tell which clock is accelerated and that Eq. 
(90) should be invalid. This disagreement has been discussed exten- 
sively in the literature; Sherwin summarizes the various arguments 
and then shows that the emitting and absorbing nuclei in a Mossbauer 
experiment play the roles of the two clocks. Assume the absorber to 
be at a very low temperature, so that v =* 0; it then plays the role of 
the stationary clock. The source at high temperature easily possesses 
rms velocities <v 2 > 1/2 comparable to that of a jet plane, and it plays 
the role of the clock moving in a space craft. According to Eq. (86) or 
(87), the emission line indeed has a lower energy than the absorption 
line. In other words, the traveling clock goes slower than the station- 
ary clock, Eq. (90) is justified, and the science fiction writers can con- 
tinue using it. 



5-5 FREQUENCY AND PHASE MODULATION 

Two methods for slightly shifting the energy of resonant gamma 
rays will be discussed in this section. These methods are not only 
useful to shift energies and hence can be employed as velocity drives 
(Sec. 3-43), but they also serve as tools for other investigations. 

The first of these methods can be called "frequency modulation of 
gamma rays." In Sec. 2-3, it was shown that if one moves the source 
of an electromagnetic wave [exp (iu> t)] with a simple harmonic mo- 
tion, x(0) sin £2t, sidebands appear with frequencies u> ±ft, w ±2J2, .... 
Ruby and Bolef (Reprint I960) have observed these sidebands by fix- 
ing a source of Fe 57 on a quartz transducer and vibrating it with a 
frequency of 20 Mc/sec. The theory borrowed from Sec. 2-3 is clas- 
sical; a quantum mechanical treatment has been given by Abragam 
(1960). 

A phase modulation of the 14.4-kev gamma ray of Fe 57 has been 
observed by Grodzins and Phillips (Reprint 1961). If a medium with 
a refractive index n and a length L is placed in the path of a resonant 
gamma ray of frequency u) , a phase shift 

$ = (1 - n)o) (L/c) (91) 

results. If either the length L or the index of refraction n is a func- 
tion of time, and if the medium is placed between source and absorber, 
then the time -dependent part of the wave function of the photon at the 
absorber is given by 



66 THE MOSSBAUER EFFECT 



f(t) = exp[-iw t + i$(t) - Tt/2n] (92) 

where r is the natural line width. The frequency spectrum of this 
phase- modulated wave train is given by the absolute square of the 
Fourier transform g(ct)), 

OO 

g(w) = 1/(2tt)V2 f at exp (icot) f (t) (93) 

o 

If one modulates the phase sinusoidally, the result is similar to 
the one found in Sec. 2-3 and in the paragraph above. Sidebands ap- 
pear and the intensity in the central line of frequency oo is dimin- 
ished. 

The saw-tooth modulation of the phase shift is more interesting. 
Assuming 

*(t) = $ t/to t o >t>0 (95) 

the integral (93) can be evaluated in closed form. All practically ob- 
tainable phase shifts $ and modulation times t are such that the 
relation 

$o<u t (96) 

is easily fulfilled. In the approximation (96), the frequency spectrum 
is given by 

«•> = i ^ i 2 s [co-. (i-v 2 : o gp + (r/2 B ) 2 (97) 

The width of the original line of energy E is unchanged, but the line 
is shifted by an amount AE given by 

AE/E = $ /co t ( 98 ) 

If the phase modulation also modulates the intensity I of the photons, 
for instance, if the photons traverse material with a linear absorption 
coefficient ju and a thickness varying according to L = L t/t so that 
I(t) = Iq exp(-juL t/t ), then the line width T in Eq. (97) must be re- 
placed by 

r'=r + ^L /t (99) 

5-6 THE UNCERTAINTY RELATION BETWEEN ENERGY AND 
TIME 

A linear harmonic oscillator serves as a simple model for a clas- 
sical radiation source. 6 If undamped, it will emit an infinitely long 



GENERAL PHYSICS 67 

wave train of sharp frequency w . If the oscillator is damped, the am- 
plitude f will decrease in time. If the damping is small, this decrease 
will be exponential, 

f(t) = f (0) exp [-iw t - (l/2)yt] (100) 

The intensity averaged over one period as a function of time is given 
by the absolute square of f(t), 

I(t) = I(0)exp(-yt) (101) 

The mean life t of this exponential decay is equal to 1/y. The fre- 
quency of such a wave train will no longer be sharp, but will be given 
by the absolute square of the Fourier transform [Eq. (93)] of f(t), 

i(w) = ^ <„ - S + m* (102) 

Equation (102) represents a Lorentz line with full width at half-height 
y centered at co . Decay time r and line width y satisfy the relation 

T-y=l (103) 

It is clear from these arguments that the line will invariably broaden 
if the wave train is shortened. Even if the decay I(t) is not exponen- 
tial, the relation (103) will yield the right order of magnitude for the 
line width y. 

In quantum theory, the expression (103) is replaced by the uncer- 
tainty relation (34). The arguments which show that a shortening of 
the time t results in a broadening of the line width y are cast in a 
different language. Rather than speaking about a wave train, one dis- 
cusses the possibility of measuring a given separation in energy AE 
within a given time T (Sec. 2-43). The emission process and its ob- 
servation can no longer be separated; both must be incorporated in 
the quantum mechanical treatment. The relevancy of Eq. (34) to these 
problems and its detailed interpretation have led to many arguments 
and discussions. One of the exciting applications of the Mossbauer 
effect is the demonstration of the correctness of the ideas leading to, 
and expressed by, Eq. (34). 

The corresponding experiments can be performed best with the 
nuclide Fe 57 (Fig. 4-1). The 123-kev gamma ray leading from the 
second to the first excited state is used to signal the time t = 0, 
i.e., the time when the first excited state of Fe 57 is populated. The 
14.4-kev gamma ray then can be observed in delayed coincidence with 
this "signal" gamma ray. By utilizing a variable delay and a resolv- 
ing time which is short compared to the mean life of the 14.4-kev pho- 
tons, one can study the shape of the Mossbauer line as a function of 



68 



THE MOSSBAUER EFFECT 



absorption 




Fig. 5-1 



cm/sec 



Doppler velocity 



The oscillatory behavior of the Fe 57 absorption 
line when observed with delayed coincidences. 
[Adapted from Wu and co-workers (I960).] 
Lifetime of the 14.4-kev state in Fe 57 r ^ 1.4 
x 10" 7 sec; delay time =* (l/2)r, resolving time 
=*0.1r. 



the time elapsed between the population of the level and the observa- 
tion of the photon. 

Such experiments were first performed by the Argonne group (Hol- 
land 1960, Reprint Lynch 1960), and later repeated by Wu and collab- 
orators (I960). Hamermesh formulated a classical theory of the time- 
dependent effects (Reprint Lynch 1960). Harris (1961) performed a 
quantum mechanical calculation and found complete agreement with 
the classical theory. 

Theory and experiment both yield the following results: For delay 
times short compared to the mean life, the Mdssbauer line is broad- 
ened in agreement with the expressions (34) and (103). This broaden- 
ing can be seen clearly in Fig. 10 of the Reprint (Lynch 1960). With 
increasing delay time, the line narrows and develops a damped oscil- 
latory behavior as shown in Fig. 5-1. For times long compared to the 
mean life, the oscillations are rapid and the central line becomes nar- 
rower than the natural line width. For most delay times, there exist 



GENERAL PHYSICS 69 

energies close to the resonance energy where the number of gamma 
rays seen by the detector is greater than if the absorber were not 
present; the absorber produces a time bunching of photons. This 
''photon excess" is shaded in Fig. 5-1. 

None of these results is new in the sense that it could not have been 
calculated years ago. However, the experimental observation of these 
effects has stimulated thinking about elementary radiation theory, and 
it allows a convincing demonstration of the uncertainty relation be- 
tween time and energy. 

Hack and Hamermesh (Reprint 1961) have treated another applica- 
tion of the Mossbauer effect to line- shape problems. Assume that a 
constant magnetic field splits the initial state of a gamma transition 
into components, separated by an energy A [Eq. (66)] . If one now ap- 
plies an external radiofrequency field of frequency co r f , one expects 
changes in the radiation pattern of the gamma ray at resonance, Hu> r f 
= A. It may be possible to detect these changes using the Mossbauer 
effect. 



5-7 RECOILLESS RAYLEIGH SCATTERING 

Consider a beam of monochromatic X rays of energy E and mo- 
mentum k which is scattered by a crystal. The scattered beam of mo- 
mentum k' contains a modified and an unmodified line. 68 The modi- 
fied line, shifted to lower energies and considerably wider than the 
incident beam, is due to Compton effect and will not be considered 
further. The unmodified line is due to Rayleigh scattering: The inci- 
dent photon of energy E is absorbed by a bound electron that is vir- 
tually excited to a higher discrete level or to the continuum. A photon 
of the same energy E is then reemitted while the electron returns to 
its original state. 29 ' 69 ' 70 The small fraction of the unmodified line due 
to nuclear Thomson scattering will be neglected. 

With the energy resolution available in ordinary X-ray experiments, 
the unmodified line appears to be essentially identical to the incident 
line. Actually, however, it consists of a coherent (elastic) and an inco- 
herent (inelastic) part. Both parts transfer the entire recoil momen- 
tum fi(k - k') to the lattice as a whole. The incoherent beam also ex- 
changes energy with the lattice by emitting or absorbing phonons. The 
energies involved in this exchange are of the order of 0.1 ev, or less. 



68 R. B. Leighton, "Principles of Modern Physics," McGraw-Hill, 
New York, 1959, p. 434. 

69 J. M. Jauch and F. Rohrlich, "The Theory of Photons and Elec- 
trons," Addison-Wesley, Reading, Mass., 1955, p. 387. 

70 G. W. Grodstein, "X-Ray Attenuation Coefficients from 10 kev 
to 100 Mev," Natl. Bur. Standards Circ. 583, 1957. 



70 THE MOSSBAUER EFFECT 

Since the widths of X rays are of the order of 1 ev, these phonon proc- 
esses simply lead to a line widening and a small line shift, but the in- 
coherent part cannot be separated from the coherent part by conven- 
tional X-ray spectrometers. The coherent or elastic part, which is 
now often called the recoilless part, leaves the lattice in its initial 
state and has exactly the same energy E as the primary beam. It is 
this coherent part of the unmodified line which is the tool of X-ray 
diffraction. 

The entire unmodified line, coherent and incoherent, is due to Ray- 
leigh scattering; the electronic state of the scatterer is not changed. 
To calculate the coherent part, one must first find the total Rayleigh 
scattering. Contributions arise from two sources— from the different 
electrons in the same atom and from electrons in different atoms. 
These contributions are partially coherent, and one must combine the 
amplitudes rather than the intensities due to the various electrons. 71 
The scattering from the different electrons of the same atom is de- 
scribed by a form factor; the scattering from different atoms is strong 
only at certain angles, the Bragg angles, and leads to the Bragg re- 
flection, used for the determination of the structure of crystals. 

Once one has calculated the total Rayleigh scattering, the coherent 
part of the unmodified line is found by taking into account the thermal 
motion of the atoms. This thermal motion gives rise to the emission 
and absorption of phonons and thus introduces energy shifts and inco- 
herence. The fraction of scattering events that occurs without energy 
loss is calculated similarly to the procedure outlined in Sec. 2-5, and 
it is given by the Debye-Waller factor [Eq. (54)]. (See also Sec. 6-22.) 

Before the discovery of the Mossbauer effect, the coherent beam 
was distinguished from the incoherent one by studying the intensity of 
Bragg reflections as a function of temperature. Tzara and Barloutaud 
(Reprint 1960) have shown that the Mossbauer effect permits a direct 
observation of the recoilless Rayleigh scattering. Resonant gamma 
rays, for instance, from a Sn 119 r an Fe 57 source, are scattered 
from a crystal that does not contain resonantly absorbing nuclei. The 
secondary beam k' contains a fraction of gamma rays that still pos- 
sess the initial energy E. They can be distinguished from the inco- 
herent background with a resonant absorber in front of the detector. 
By measuring the velocity spectrum, i.e., by moving the resonant ab- 
sorber with a velocity v with respect to the stationary source and 
scatterer, one can trace out the form of the coherent line in a manner 
similar to the investigation of the transmission line shown in Fig. 3-1. 



71 A small problem in semantics arises here. Both the coherent 
and incoherent part of the unmodified line must be calculated, taking 
into account the coherence among the various atoms and electrons. 
The coherence among the atoms is then partly destroyed by tempera- 
ture motion, and this gives rise to the incoherent part. 



GENERAL PHYSICS 71 

The possibility of cleanly separating the coherent from the incoherent 
line is of course due to the fact that the gamma rays used in these ex- 
periments possess a line width of about 10" 7 ev or less. This width is 
not changed by the coherent scattering, whereas the incoherent line is 
shifted and is smeared out over an energy of about 0.1 ev. 

5-8 COHERENCE AND INTERFERENCE 

The theory of coherence and interference involving Mossbauer scat- 
tering has been treated in a number of publications (Reprint Kastler 
1960, Reprint Tzara 1961, Podgoretskii 1960, Lipkin 1961, Moon 1961, 
Raghavan 1961, Trammel 1961, Tassie 72 ). Relevant experiments have 
been performed by Black and Moon (Reprint 1960) and by Major (1961). 
More recent results have been reported by Black. 73 

The concept of coherence originated in classical optics. Even though 
it is basically not a difficult concept, its application is often confusing, 
and errors can arise because the physical situation and the meaning of 
coherence have not been clearly defined. 74 ' 75 Here, a simple example 
should suffice. Consider two radiation sources, each emitting waves 
of frequency w with amplitudes a= |a| exp[io?(t)] and b= |b| exp[i/3(t)], 
where a and are real. The intensity I(t) observed at a given point 
r in space and averaged over one period will be the absolute square 
of the sum of the amplitudes 

I(r,t) = |a + b| 2 = |a| 2 + |b| 2 + I int (104) 

where the interference term I^ nt is given by 

I int = 2 |a| |b| cos (a - j3) = 2 Re(a* b) (105) 

The two waves a and b are said to be incoherent at the point r if 
the intensity I(r,t) is the sum of |a| 2 and |b| 2 , i.e., if the interfer- 
ence term vanishes. The waves will be perfectly coherent if the 
phases of the two bear a definite relationship to each other and 
slightly coherent if there exists a small correlation between their 
phases. 

In quantum theory the discussion of coherence can be made along 
the same lines: If the probability of finding photons from two sources 
at a given point is not equal to the sum of the probabilities, one says 
that the two waves possess a certain amount of coherence. This def- 
inition applies particularly to the scattering of one photon from two 



72 L. J. Tassie, unpublished report. 

73 P. J. Black, in "Proceedings of the Second Mossbauer Confer- 
ence, " Paris, Sept. 13-16, 1961, A. Schoen and D. M. J. Compton 
(eds.), Wiley, New York, tentative publication Spring 1962. 

74 A. T. Forrester, Am. J. Phys., 24, 192 (1956). 

75 E.L. O'Neill and L.C.Bradley, Phys. Today, 14 (6), 28 (1961). 



72 THE MOSSBAUER EFFECT 

scattering centers (see Ref. 6, pp. 192-194, 202; and Ref. 23, pp. 133- 
137). The scattered waves can be coherent; i.e., they can interfere 
with each other provided there is a fixed phase relation between them. 
The coherence is destroyed, however, if one determines by a meas- 
urement, for instance of the recoil of one center, from which center 
the photon has been scattered. Such an investigation with fixed phase 
relation is analogous to the famous optical double- slit experiment. 

In order to discuss specifically the processes that can interfere 
with Mossbauer scattering, consider a photon of momentum lik which 
strikes a resonant scatterer and excites a nucleus to its first excited 
state by recoilless absorption. The reemission can result in a photon 
of momentum ftk' or in a conversion electron. If one detects photons, 
Rayleigh scattering by the atomic electrons of one atom or by the 
electrons of many atoms can be coherent with the Mossbauer scatter- 
ing. If one detects electrons, the photo effect and the internal conver- 
sion can interfere. 72 If the resonant nuclei in the absorber are abun- 
dant enough, scattering from the different nuclei can be coherent. In 
all these examples, coherence exists only if one does not determine 
where the scattering occurs. If, for instance, a hyperfine interaction 
is present at the resonant nuclei, then its observation shows that res- 
onance scattering has occurred and all coherence with Rayleigh scat- 
tering is destroyed. 

The equations describing the cross sections in the presence of co- 
herent processes are given by Tzara (Reprint 1961). Here some of 
the essential facts are summarized. If one observes all processes 
leading to scattering in the direction k', without discriminating against 
those which leave the lattice in an excited state, the cross section is 
given by (Lipkin 1961) 

a oc [|R|2 + | M | 2 f(k) + 2C Re (R*M) f(k)] (106) 

where R is the probability amplitude for Rayleigh scattering plus nu- 
clear Thomson scattering, M is the probability amplitude for the 
Mossbauer scattering, and f(k) is the Lamb- Mossbauer factor, Eq. 
(49) or (56). C is a factor describing the degree of coherence be- 
tween the processes; it is independent of the lattice (Moon 1961). 

If one uses a resonant detector to observe only those scattering 
events which leave the lattice in its initial state, one finds 

a cc[|R|*F'(k - k') + |M| 2 f 2 (k) + 2C Re(R*M)F 1/2 (k-k') f(k)] 

(107) 

where k-k' is the momentum transferred to the lattice and 
F(k - k') is the corresponding Debye-Waller factor, Eq. (54). 

A few remarks about the interpretation and the application of these 
equations are in order. 



GENERAL PHYSICS 73 

1. Equation (106) and Eq. (107) show that one must choose R and 
M of about equal magnitude in order to make the interference term 
as large as possible compared with the other terms. 

2. The dependence of the interference term on the Debye-Waller 
factor F (k - k') and on the Lamb- Mossbauer factor is clearly exhib- 
ited by the two equations, (106) and (107), and needs no further dis- 
cussion. 

3. The interference between Mossbauer scattering and Rayleigh 
scattering has been observed experimentally by Black and Moon 73 
(Reprint 1960). In order to get a large effect, they selected scattering 
angles such that strong Bragg reflections occurred (compare Sec. 
5-7). The interference term changes sign when one changes the en- 
ergy from slightly above the Mossbauer resonance to slightly below, 
and vanishes at exact resonance. This effect also is expected (Tzara 
1961, Moon 1961). 

4. Campbell and Bernstein 76 have observed interference between 
the Mossbauer scattering and totally reflected gamma rays. This 
technique permits the study of the chemical and magnetic environment 
of those nuclei which lie in a very thin layer close to the surface. 

5. One possible application of interference effects may well be the 
observation of a weak Mossbauer line in a strong background, since 
the interference term depends on M rather than on |M| 2 (Moon 1961). 

6. Another possible application is to the determination of phases in 
crystallography (Raghavan). 

5-9 POLARIZATION 77 

The optical Zeeman effect has been extremely important for the 
understanding of atomic structure. In nuclear physics, prior to the 
discovery of the Mossbauer effect, analogous experiments were im- 
possible. It is true that many polarization experiments have been per- 
formed with nuclear gamma rays, but these experiments have been 
difficult and cumbersome. Moreover, they always had to be performed 
on components unresolved in energy, and the information gathered 
about the polarization of individual components was rather indirect. 

The discovery of the Mossbauer effect has changed this situation. 
Particularly with the 14.4-kev gamma ray emitted by Fe 57 , the nuclear 
Zeeman effect can be observed easily and in detail (see Sec. 4-3). The 
individual components of the emitted gamma-ray lines are widely sep- 
arated compared to the natural line width. The 93-kev gamma ray in 
Zn 67 displays a Zeeman splitting of many line widths in fields as small 
as 100 gauss (Reprint Craig 1960). 



76 E. C. Campbell and S. Bernstein, Bull. Am. Phys. Soc, 6, 443 (1961), 
77 This section follows closely some parts of a Los Alamos publica- 
tion (Frauenfelder 1962). 



74 THE MOSSBAUER EFFECT 

Once the Zeeman components are separated clearly, the determina- 
tion of their state of polarization becomes the next step. Experiments 
with the plane polarization of the Fe 57 gamma rays were first per- 
formed by the Argonne group in order to complete the understanding 
of the Mossbauer spectrum of Fe 57 embedded in iron (Reprint Hanna 
1960). The gamma-ray polarization has also been taken into account 
by Wegener and Obenshain in order to explain the shape of the lines 
observed in Ni 61 (Wegener 1961). The elliptical polarization was first 
used to simplify the investigation of the complicated spectra that ap- 
pear when Fe 57 is embedded in CoPd (Reprint Nagle 1960). 

Polarization measurements on resolved gamma-ray components 
can be a very effective tool for unraveling complex spectra, for re- 
ducing the number of lines in complicated spectra, and for finding the 
direction of the internal magnetic field inside magnetic domains in 
ferromagnets, ferrimagnets, antiferromagnets, and possibly super- 
conducting ferromagnets. 

A complete theory of the polarization of gamma rays emitted and 
absorbed in the Mossbauer effect has been worked out by Visscher 
(Frauenf elder 1962), and this paper should be consulted for details. 
In the present section, a few remarks and the description of a simple 
experiment show the ease with which the elliptical polarization can be 
demonstrated in the Mossbauer effect. A similar very simple exper- 
iment showing the linear polarization of the Fe 57 gamma rays has 
been performed by the Argonne group (Perlow 1960). 

The levels of Fe 57 , embedded in ferromagnetic iron metal and 
placed in an external magnetic field, are shown in Fig. 2 of the Re- 
print (Hanna I960). 78 The radiation pattern emitted by Fe 57 will gen- 
erally consist of six lines, as shown in the same figure. However, if 
one observes the pattern along the axis of the external magnetic field, 
the transition Am = is completely forbidden, and only four compo- 
nents appear. To discuss the polarization of the four components, one 
notes that a gamma ray is called right circularly polarized if its spin 
lies in the direction of motion. (This convention is opposite to the one 
used in optical spectroscopy.) Consider first the highest energy com- 
ponent, namely, the transition -3/2 to -1/2. The z component of an- 
gular momentum must be conserved, and any gamma ray correspond- 
ing to this component hence must have a z component of angular 
momentum of -1. If this gamma ray is emitted along the +z direc- 
tion, its spin is antiparallel to its momentum, and it is left circularly 



78 The labeling of the magnetic sublevels of Fe 57 has led to some 
confusion. The magnetic moment of the excited state is negative; the 
internal field in iron metal is opposite to the externally applied mag- 
netic field. If one uses the convention that the external field is di- 
rected along the +z axis, then the m = -3/2 state possesses the high- 
est energy, as shown in Fig. 2 of the Reprint (Hanna, 1960). 



GENERAL PHYSICS 



75 



Fe 57 source 



Fe 57 absorber 



counter 




Fig. 5-2 A simple experiment to demonstrate the circu- 
lar polarization of the 14.4-kev gamma rays 
from Fe 57 . The counting rate with parallel fields 
is much larger than with antiparallel fields at 
source and absorber. 



polarized. If it is emitted in the -z direction, spin and momentum are 
parallel and the photon is right circularly polarized. Similarly, one 
can see that the other components are also circularly polarized. 

The primitive experimental setup that serves to demonstrate the 
elliptical polarization of the Fe 57 gamma rays emitted closely paral- 
lel to the field axis is shown in Fig. 5-2. (Emission completely par- 
allel is hard to achieve experimentally.) Source and absorber consist 
of iron metal and both are placed in solenoids. If the fields in the two 
solenoids are parallel, the absorption at the central peak, i.e., at 
zero relative velocity between source and absorber, is very much 
larger than that for antiparallel fields. The explanation is obvious. 
At small angles with the field axis, the emitted and the absorbed pho- 
tons are nearly circularly polarized. In parallel fields, the right cir- 
cular components of the source have the same energy as the right 
circular absorption levels, and the left circular emission components 
have the same energy as the corresponding left circular absorption 
levels and the absorption is large. Reversing the field, for instance, 
at the source, changes right into left circular and vice versa. The 
energies for equal polarization in source and absorber no longer are 
the same, and the absorption at zero relative velocity is small. 




SOLID-STATE PHYSICS 



6-1 SURVEY 



In Chapter 4 it was pointed out that a striking similarity exists be- 
tween the histories of nuclear magnetic resonance and the Mossbauer 
effect. Originally both effects belonged to nuclear physics, but the 
solid-state applications soon dominated both fields. The reason is 
clear: The number of nuclear properties that can be investigated is 
limited, but the number of solid-state parameters that can be varied 
and studied is very large. It seems at the present time as if most of 
the future investigations involving the Mossbauer effect will be in 
solid-state physics. In the present chapter a brief sketch of some of 
the major applications will be given. A more detailed discussion of 
recent work can be found in the "Proceedings of the Second Moss- 
bauer Conference" (see Sec. 7-3). 

The quantities that can be measured in a Mossbauer experiment, 
namely, the Lamb- Mossbauer factors, the line shape, the line splitting, 
and the line shift, have been discussed in Chapter 3. The fact that these 
quantities can be determined under a wide variety of conditions and 
with a wide range of parameters makes the recoilless gamma-ray 
emission and absorption such a powerful tool in solid-state physics. 

The choice of parameters to be varied determines which solid-state 
properties will be investigated. One can, first of all, embed the reso- 
nant nuclei in the source or in the absorber. The host can be of the 
same chemical composition as the radioactive material, or it can be 
different. The resonant nuclei can be interstitial or substitutional im- 
purities; their concentration can be varied over wide ranges. The host 
lattice can be well annealed or it can be strained. Its temperature can 
be varied or the external pressure can be changed. External magnetic 
or electric fields can be applied. The emitted or absorbed radiation 
and its characteristics can be studied at various angles with respect 

76 



SOLID-STATE PHYSICS 77 

to internal or external fields or nonisotropic lattice properties. The 
Mossbauer effect can be measured in transmission or in scattering; 
one can observe gamma rays, conversion electrons, or X rays follow- 
ing internal conversion. Interference effects can be used. The time 
dependence of various properties can be studied either by means of a 
delay technique or by embedding isotopes with different excited-state 
lifetimes into the same host lattice. 

Four typical applications of the Mossbauer effect to solid-state 
problems will be discussed in the next sections. These four, lattice 
properties (Sec. 6-2), internal fields (Sec. 6-3), impurities and im- 
perfections (Sec. 6-4), and low -temperature topics (Sec. 6-5), do not 
exhaust all the possibilities. The Mossbauer effect, for instance, has 
also been used to get information about atomic wave functions at the 
nucleus (Reprint Walker 1961) and to determine the index of refrac- 
tion of lucite (Reprint Grodzins 1961). These are but two examples of 
investigations that are not contained in Sees. 6-2 to 6-5. Very likely 
the Mossbauer effect will also be utilized to study surfaces, liquids, 
and some problems in chemistry and biophysics. 

The Mossbauer effect is the youngest of the nuclear guests in solid- 
state physics. It fits in well indeed with the older ones, complementing 
them in some areas and allowing checks in others. Table 6-1 lists the 
major nuclear tools in solid-state physics and indicates the areas of 
information which they have in common with the Mossbauer effect. 

Table 6-1 Nuclear Tools in Solid- State Physics 

Topics in common with the 

Tool Mossbauer effect Refs. 

X-ray diffraction Lattice dynamics 18, 27-31, 79 

Neutron diffraction Lattice dynamics Reprint Lamb 1939 

Magnetic properties 80-83 

Nuclear magnetic Internal fields 84-89 

resonance Atomic wave functions 

at the nucleus 

Angular correlation] Internal fields 90-93 

Oriented nuclei Aftereffects of radio- 94, 95 

active decays 



79 J. Bouman in ' 'Encyclopedia of Physics," Springer, Berlin, 1957, 
Vol. 32. 

80 C. G. Shull and E. O. Wollan in F. Seitz and D. Turnbull (eds.), 
"Solid State Physics," Academic, New York, 1956, Vol. 2, p. 138. 

81 G. R. Ringo in "Encyclopedia of Physics," Springer, Berlin, 1957, 
Vol. 32. 

82 E. Amaldi in "Encyclopedia of Physics," Springer, Berlin, 1959, 
Vol. 38/2. 



78 THE M5SSBAUER EFFECT 



6-2 LATTICE PROPERTIES 

Boyle recently divided the Mossbauer experiments to determine 
lattice properties into three classes: difficult and uninteresting, very 
difficult, impossible. 96 Since even impossible experiments sometimes 
succeed, a few of the applications of the Mossbauer effect to lattice 
problems are sketched in the present section. 

6-21 The Lamb- Mossbauer Factor f 

This factor, Eq. (49), can be determined by measuring the fraction 
of gamma rays emitted or absorbed without energy loss as a function 
of the temperature T. From f(T) one finds the mean- square devia- 
tion <X 2 > by using Eq. (56). Further evaluation depends on the model 
one selects to describe the solid. Usually one determines a Debye 
temperature with the help of Eqs. (52) and (54) (see also Ref . 97). 



83 L. S. Kothari and K. S. Singwi in F.Seitz and D. Turnbull (eds.), 
"Solid State Physics," Academic, New York, 1959, Vol. 8, p. 109. 

84 E. R.Andrew, "Nuclear Magnetic Resonance," Cambridge Univer- 
sity Press, New York, 1955. 

85 G.E.Pake in F.Seitz and D. Turnbull (eds.), "Solid State Physics," 
Academic, New York, 1956, Vol. 2, p. 1. 

86 W. D. Knight in F. Seitz and D. Turnbull (eds.), "Solid State 
Physics," Academic, New York, 1956, Vol. 2, p. 93. 

87 M.H.Cohen and F.Reif in F.Seitz and D. Turnbull (eds.), "Solid 
State Physics," Academic, New York, 1957, Vol. 5, p. 322. 

88 T. P. Das and E. L. Hahn, "Nuclear Quadrupole Resonance Spec- 
troscopy," Supplement 1 to "Solid State Physics," Academic, New 
York, 1958. 

89 A. Abragam, "The Principles of Nuclear Magnetism," Oxford 
University Press, New York, 1961. 

90 H. Frauenfelder in K. Siegbahn (ed.), "Beta- and Gamma-Ray 
Spectroscopy," North- Holland, Amsterdam, 1955. 

91 R. M. Steffen, Advances in Phys., 4, 293 (1955). 

92 S. Devons and L. J. B. Goldfarb in "Encyclopedia of Physics," 
Springer, Berlin, 1957, Vol. 42. 

93 E.Heer and T.B.Novey in F.Seitz and D. Turnbull (eds.), "Solid 
State Physics," Academic, New York, 1959, Vol. 9, p. 199. 

94 R. J. Blin-Stoyle, M. A. Grace, and H. Halban in K. Siegbahn (ed.), 
"Beta- and Gamma-Ray Spectroscopy," North- Holland, Amsterdam, 
1955. 

95 R. J. Blin-Stoyle and M. A. Grace in "Encyclopedia of Physics," 
Springer, Berlin, 1957, Vol. 42. 

96 A. J. F. Boyle in "Proceedings of the Second Mossbauer Confer- 
ence," Paris, Sept. 13-16, 1961, A. Schoen and D. M. J. Compton(eds.), 
Wiley, New York, tentative publication Spring 1962. 



SOLID-STATE PHYSICS 79 

Boyle et al. (Reprint 1961) have shown that one can also get infor- 
mation about anharmonic terms from f(T). The reader is referred to 
their paper for a discussion of the procedure and of the approxima- 
tions involved in the evaluation. 

Singwi and Sjolander (Reprint 1960) pointed out that in an anisotro- 
pic solid, such as graphite, one should expect a dependence of f and 
of the line width on the direction of emission. Corresponding calcula- 
tions by Kagan (1961) corroborate this statement, and an anisotropic 
behavior has actually been found for Fe 57 embedded in graphite. 98 ' 99 

Two more possible experiments should be mentioned here. It is of 
interest to determine whether f(T) will follow the Debye behavior at 
very low temperatures or whether deviations will occur. Investiga- 
tions in molecular crystals should reveal how additional degrees of 
freedom affect f(T). 

6-22 The Debye- Waller Factor 

This factor, F (k - k') , can be found with the help of the recoilless 
Rayleigh scattering (Sec. 5-7). A few words about the difference be- 
tween the Lamb- Mossbauer factor f(k) and the Debye- Waller factor 
F(k - k') are in order here (Reprint Tzara 1961, Lipkin 1961, Tram- 
mell 1962). Consider a process in which an incoming beam, neutrons 
or photons, of momentum tlk, is scattered by a solid into a momentum 
state nk' . If the scattering process is nonresonant , then the time de- 
lay (collision time, phase shift) between incoming and outgoing wave 
is of the order of, or less than, v/d, where v is the beam velocity 
and d the linear dimension of the scatterer. 100 This delay time is 
much shorter than the characteristic lattice time -R/k0: An X-ray 
wave front moves across an atom in a time of the order of 10" 18 sec, 
whereas typical lattice vibration times are longer than 10" 14 sec. In 
scattering of atomic X rays, the duration of the entire wave train is 
of the order of 10" 15 sec or less and hence also short compared to the 
lattice vibration time. In a crude way of speaking, the entire scatter- 



97 W. Marshall in "Proceedings of the Second Mossbauer Confer- 
ence," Paris, Sept. 13-16, 1961, A. Schoen and D.M.J. Compton(eds.), 
Wiley, New York, tentative publication Spring 1962. 

98 D. E. Nagle in "Proceedings of the Second Mossbauer Confer- 
ence," Paris, Sept. 13-16, 1961, A. Schoen and D. M. J. Compton(eds.), 
Wiley, New York, tentative publication Spring 1962. 

99 H. Pollak, M. DeCoster,and S. Amelincks in "Proceedings of the 
Second Mossbauer Conference," Paris, Sept. 13-16, 1961, A. Schoen and 
D. M. J. Compton (eds .), Wiley, New York, tentative publication Spring 
1962. 

100 D. Bohm, "Quantum Theory," Prentice- Hall, Englewood Cliffs, 
N.J., 1951, p. 261. 



80 THE MOSSBAUER EFFECT 

ing process occurs on atoms that move very little during the scatter- 
ing process. The entire momentum h(k - k') is transferred to the 
lattice during such a nonresonant collision. The fraction of X rays 
scattered without energy loss is found by averaging over the various 
positions of the scattering atom. This calculation leads to the Debye- 
Waller factor, 

F(k- k') = exp -<[(k- k')-X] 2 > (108) 

where X is the coordinate vector of the cm. of the decaying nucleus 
(Fig. 2-3). The fraction F of recoilless nonresonant scattering de- 
pends on the scattering angle 2cp; it increases with decreasing angle 
<p, i.e., with decreasing momentum transfer ft(k - k'). Since the 
scattering is elastic, |k - k'| = 2k sin </?, and Eq. (108) is thus identi- 
cal with Eq. (54). 

The nonresonant scattering (Rayleigh scattering) with nuclear 
gamma rays from a resonant source is also described by Eq. (108). 
Even though the duration of the wave train is long compared to the lat- 
tice vibration time, the time delay is short. At each instant of time 
the conditions implied in Eq. (108) are fulfilled, and the averaging over 
the motion of the scattering atoms again leads to Eq. (108). The inten- 
sity of the recoilless Rayleigh scattering thus measures the Debye- 
Waller factor F(k- k'). 

The resonant scattering leads to a different expression. Here, the 
time delay between incident and outgoing wave front is of the order of 
r = "fi/r, i.e., of the order of the lifetime of the resonant state. 100 In 
Mdssbauer scattering this lifetime is much longer than the lattice vi- 
bration times, and there is negligible correlation between the posi- 
tions of absorption and subsequent reemission. The fraction f s of 
gamma rays scattered without energy loss is then given by the prod- 
uct of two Lamb- Mdssbauer factors, 

f s = f(k)f(k') (109) 



6-23 Phonon Spectra 

It has repeatedly been suggested that a detailed measurement of 
the energy spectrum of the gamma rays in the Mdssbauer effect, par- 
ticularly the observation of the one phonon exchange, will yield valu- 
able information about phonon spectra (Reprint Visscher 1960, Petz- 
old 1961). Unfortunately such experiments are extremely difficult 
since the energy range to be covered is of the order of 10" 2 ev and 
hence much wider than even very wide lines. Neutrons are much bet- 
ter suited for such investigations (Reprint Singwi 1960), except for 
elements with very large capture cross sections, such as He 3 . 



SOLID-STATE PHYSICS 81 



6-24 Diffusion 

Diffusion should lead to a broadening of the emission and absorp- 
tion lines close to the melting point (Reprint Singwi 1960). It is very 
likely that such a broadening can be observed experimentally. 

6-25 Lattice Specific Heats 

These can be determined by the observation of the second-order 
Doppler effect, as pointed out in Sec. 5-4, Eq. (88). The main prob- 
lem is the unambiguous separation of the various temperature- 
dependent shifts. 

6-26 Pressure Effects 

Increasing the ambient pressure p on a sample will increase the 
Lamb-Mossbauer factor f (Hanks 1961) and will shift the resonance 
line 101 (Pound 1961). 

To calculate the pressure dependence f(p) of the Lamb-Mossbauer 
factor, it is easiest to assume that the Debye model (Sees. 2-2 and 
2-52) applies so that f - f(e). Then one has 

af/ap = (af/ae) (ae/av)(av/ap) (no) 

The dependence of the Debye temperature on the volume V is 
given by the Griineisen relation 21 : 

3 In 0/a In V= -y (111) 

Neglecting the volume dependence of the Griineisen constant y, Eq. 
(Ill) can be written 

de/dv= -ye/v . (112) 

The volume V depends on the pressure p through the compressibility 

3V/ap = -KV (113) 

For simplicity assume temperatures low enough so that Eq. (33) 
holds: f = exp(-3R/2k0). Then one finds with Eqs. (33) and (110) to 
(113), after integration, 

f(p) = f(p = 0) exp [(3R/2k0) /cyp] = exp[-(3R/2k0)(l-/cyp)] (114) 



101 R.V. Pound in "Proceedings of the Second Mossbauer Confer- 
ence," Paris, Sept. 13-16, 1961, A. Schoen and D. M. J. Compton (eds.), 
Wiley, New York, tentative publication Spring 1962. 



82 THE MOSSBAUER EFFECT 

The Griineisen constant y is of the order of 2 (Table 10.3 in Ref. 21), 
the volume compressibility k varies from about 3 x 10~ 7 atm" 1 for W 
to about 5 x 10" 5 atm" 1 for Cs. As pointed out by Hanks (1961), it 
should thus be possible to observe the pressure dependence of f. It 
should even be feasible to detect a pressure- induced recoilless emis- 
sion or absorption in substances where f is too small to be seen at 
normal pressures. 

The second manifestation of a change of the ambient pressure, 
namely, a shift of the Mossbauer line, is mainly due to two contribu- 
tions. One is caused by a change with pressure of the electronic wave 
functions at the nucleus; this term can be called the volume depend- 
ence of the isomeric shift (Sec. 4-4). The other contribution comes 
from the volume dependence of the internal energy of the solid; this 
term is analogous to the second-order Doppler shift (Sec. 5-4). One 
interesting difference between the second-order Doppler shift and the 
pressure shift appears: The former does not depend on the zero-point 
energy and thus goes to zero at low temperatures. The pressure shift, 
however, does depend on the zero-point energy and thus remains finite 
even at very low temperatures. 

The dependence of the Lamb- Mossbauer factor f on pressure has 
not yet been observed experimentally. The pressure shift, however, 
has been investigated by Pound 101 who found a relative shift 
(l/E)(dE/3p) = -2.6 x 10" 18 atm -1 for Fe 57 , in good agreement with his 
theoretical estimates (Pound 1961). 

6-3 INTERNAL FIELDS 

An inspection of the bibliography (Chapter 7) shows that the largest 
number of publications is devoted to the application of the Mossbauer 
effect to the study of internal fields. These investigations have al- 
ready greatly enhanced the knowledge of internal fields and it is very 
likely that the fast flow of information will continue. Despite the im- 
portance of internal field studies, the present section is short. Excel- 
lent surveys already exist, for instance in the notes by Abragam (see 
Sec. 7-1) and in the paper by Wertheim (Reprint 1961). Moreover, 
the basic ideas underlying the study of internal fields have already 
been outlined in the discussion of the nuclear moments in Sec. 4-3. 

The solid-state physicist's view of Sec. 4-3 is obvious: Instead of 
measuring nuclear g factors and nuclear quadrupole moments, his 
interest is directed to the extranuclear fields responsible for the in- 
teractions. To determine these extranuclear fields, one requires a 
knowledge of the magnitude and sign of the nuclear moments. The 
ground-state moments can usually be determined by conventional tech- 
niques. When the ground-state moments are known, the excited-state 
properties can usually be inferred from the Mossbauer pattern. Once 
the nuclear moments are known, the dependence of the internal fields 



SOLID-STATE PHYSICS 83 

on various parameters, such as crystal structure, temperature, pres- 
sure, and external fields, can be studied. 

In Sec. 6 of the reprints a number of representative papers are 
collected. These give a fair survey of the problems that have already 
been examined and the tools used in measuring and interpreting spec- 
tra. All these papers use either Fe 57 or Sn 119 as the probe. These 
two nuclides permit a tremendous amount of research, and their use 
has already led to a deeper understanding of the magnetic properties 
of solids. The field at the iron nucleus in iron metal constitutes a 
good example. The first experiments (Reprint Hanna 1960, Sec. 4 of 
the reprints) yielded a magnitude of the magnetic field |H| = 3.3xl0 5 
oersteds, which was in good agreement with the theoretical predic- 
tions. Then the Argonne group went one step farther and determined 
also the sign of H (Reprint Hanna 1960, Sec. 6 of the reprints). They 
found the internal field to be opposite to the externally applied field, 
in direct contradiction to theoretical prediction. Since then theoretical 
physicists have obtained agreement with experimental facts. 102 

In addition to studies with Fe 57 and Sn 119 , a considerable amount of 
work has been performed with rare earth nuclides, and with nuclides 
like Au 197 embedded in ferromagnetic materials (see the bibliography). 
These areas promise to be as interesting and rewarding as the work 
discussed above. 

Investigations of quadrupole effects have been performed also (Re- 
print Kistner 1960, Reprint DeBenedetti 1961, see also the bibliogra- 
phy). Quadrupole splittings have been found in many substances and a 
beginning has been made to correlate these splittings with the chemi- 
cal and physical state of the solid. The interpretation is difficult when 
magnetic and quadrupole interactions are simultaneously present. The 
splitting then depends on the orientation of the magnetic field with re- 
spect to the electric field gradient (e.g., Wertheim 1961a), and it is 
advantageous to work with single crystals. 

6-4 IMPURITIES AND IMPERFECTIONS 

Every Mossbauer experiment involves nonideal lattices. Sadly, 
little work has been done so far to systematically investigate the in- 
fluence of deviations from an ideal lattice on recoilless emission or 
absorption. A number of publications deal with internal fields at im- 
purity atoms but this aspect will not be discussed. 

6-41 Impurities 

Impurities actually form a special class of imperfections, but they 
are singled out here because each radioactive atom that emits a 



102 



R. E. Watson and A. J. Freeman, Phys. Rev., 123, 2027 (1961). 



84 THE MdSSBAUER EFFECT 



gamma ray can be considered an impurity atom. These radioactive 
atoms can be different from the normal lattice in a number of proper- 
ties, such as mass, binding, or radius; they can also be in interstitial 
positions. 

The crude models that have been considered so far (Shapiro 1961, 
Reprint Maleev 1961, Visscher 1962) predict some features which can 
be checked by experiment. Two of these predictions are particularly 
simple to describe, namely, the influence of the mass M of the im- 
purity atom and the existence of localized oscillations. Assume that 
the ideal lattice consists of atoms with a mass m and that the radio- 
active impurity atom possesses a mass M. The predominant part of 
the mean-square deviatibn <X 2 > comes from frequencies far below 
the Debye frequency o>d because the contribution of each individual 
mode, <Xi>, is proportional to 1/wi. The low-frequency modes 
have wavelengths long compared to the lattice spacing, and the neigh- 
boring atoms move predominantly in the same direction. The displace- 
ment of the impurity atom with respect to its neighbors is then small 
compared to its displacement X from the equilibrium position. The 
mean- square deviation <X 2 > of the impurity atom is in a first ap- 
proximation the same as that of the normal lattice atoms. In the 
Debye approximation, the Lamb-Mossbauer factor f is thus given by 
Eq. (53), with R = E 2 /2mc 2 . If this description is correct, then the 
fraction f should be determined by the mass m of the atoms in the 
host lattice, and not by the mass M of the impurity atom. This con- 
clusion has, however, not yet been substantiated by experiments. 

An impurity atom in an ideal lattice produces localized oscillations, 
i.e., oscillations that are large in its immediate vicinity and damp out 
quickly with increasing distance from it. 103 These localized oscilla- 
tions should give rise to individual discrete peaks in the recoil spec- 
trum, separated from the unshifted line by energies of the order of 
0.01 ev (Reprint Maleev 1961, Visscher 1962). 

6-42 Imperfections 

A second type of problem occurs when the host lattice is not ideal, 
which is the situation normally encountered in experiments. Various 
types of imperfections can then influence the Mossbauer effect. Even 
though little work has been done on this aspect, there is no doubt that 
experiments involving imperfections will play an increasingly larger 
role in future research. 



6-5 LOW TEMPERATURES 

There exist some problems which are inherently dependent on tern- 



103 A. A. Maradudin, P. Mazur, E. W. Montroll, and G. H. Weiss, 
Revs. Modern Physics, 30, 175 (1958). 



SOLID-STATE PHYSICS 85 

peratures close to, or below, the helium boiling point and which can 
be investigated with the Mossbauer effect. Two of these will be 
sketched here. 

6-51 Nuclear Orientation 

In conventional nuclear orientation experiments, 94 ' 95 the degree of 
orientation must be deduced from angular distributions of nuclear radi- 
ations. With the Mossbauer effect, this property can be determined 
more directly, at least in favorable cases. The ease and accuracy with 
which such measurements can be performed make it likely that the 
Mossbauer effect can be used as a thermometer at very low tempera- 
tures (Taylor 1962). 

As a simple example, consider a single line Co 57 —* Fe 57 source and 
an Fe 57 absorber that possesses a strong internal field H. The ground 
state of the Fe 57 nuclei in the absorber is split into two magnetic sub- 
levels, separated in energy by A = gfi H [Eq. (66)]. At a temperature T, 
the ratio of the equilibrium populations of these two sublevels is 

N(+l/2)/N(-l/2) = exp (-A/kT) = exp (-g/i H/kT) (115) 

The ratio of symmetric absorption components, for instance, 
(l/2-~3/2)/(-l/2^ -3/2), is also given by (115). At very low temper- 
atures, where A^>kT, the component originating from the higher sub- 
level is absent. In general, both components will be present; their in- 
tensity ratio is a direct measure for the Boltzmann factor (115) and 
hence yields the temperature T if g and H are known. 

An Fe 57 absorber works as a thermometer only at very low tem- 
peratures because the magnetic moment of the Fe 57 ground state is 
very small. The splitting in iron metal corresponds to a temperature 
T* = A/k = 2.2 x 10~ 3 °K; appreciable effects hence occur only in the 
millidegree range. The Los Alamos group performed an experiment 
at higher temperatures by using the much larger magnetic moment of 
the parent nucleus Co 57 (Reprint Dash 1961). 

6-52 Superconductivity 

The application of the Mossbauer effect to superconductivity is 
tempting at first glance because the energy changes detectable in the 
Mossbauer effect and the energy change per atom in superconducting 
transitions are of the same order of magnitude. A more detailed dis- 
cussion shows, however, that one should not expect sizeable effects, 
because very few electrons participate in a superconducting transition. 
Careful experiments by the Los Alamos group (Craig 1960, 1961) have 
indeed failed to find an influence of a superconducting transition on 
the Mossbauer effect. Recent experiments (Wiedemann 1961) which in- 
dicate a very small change in f, are not conclusive. 




BIBLIOGRAPHY ON THE 
MOSSBAUER EFFECT 

Compiled by E. Liischer, D. Pipkorn, and M. Runkel 
Physics Department, University of Illinois 

7-1 REVIEW ARTICLES 

A. Abragam, L'effet Mossbauer et ses applications a Petude des 
champs internes (unpublished lecture notes). 

G. N. Belozerskii and Yu. A. Nemilov, The resonance dispersion of 
y-rays in crystals, Uspekhi Fiz. Nauk, 72, 433 (1960) (in Russian); 
Soviet Physics Uspekhi, 3, 813 (1961) (in English). 

A. J. F. Boyle and H. E. Hall, Mossbauer effect, Repts. Progr. in 
Phys. (to be published in 1962). 

W. E. Burcham, Nuclear resonant scattering without recoil (Moss- 
bauer effect), Sci. Progr., 48, 630 (1960). 

E. Cotton, Emission et absorption de rayonnement gamma sans recul 
du noyau emetteur emprisonne dans un reseau cristallin (Effet 
Mossbauer), J. phys. radium, 21, 285 (1960). 

P.P. Craig, Experimental aspects and applications of the Mossbauer 
effect, in G. M. Graham and A. C. Hollis -Halle tt (eds.), "Pro- 
ceedings of the Vllth International Conference on Low Tempera- 
ture Physics," University of Toronto Press, Toronto, 1961, 
pp. 22-35. 

S. DeBenedetti, The Mossbauer effect, Sci. American, 202, 72 (1960). 



* Entries in this bibliography are arranged (a) alphabetically by 
author, (b) by added joint authors, (c) alphabetically by title. In 
cases where two or more papers by the same authors appeared in 
the same year, the second paper (alphabetically by title) is desig- 
nated a, the third b, etc. 

86 



BIBLIOGRAPHY 87 

W. E. Kock, The Mossbauer radiation, Science, 131, 1588 (1960). 
I. Y. Krause and G. Liiders, Kernresonanzabsorption mit eingefro- 

renem Ruckstoss, Naturwissenschaften, 47, 532 (1960). 
H. Lustig, The Mossbauer effect, Am. J. Phys., 29, 1 (1961). 
C. Moller, The Mossbauer effect, pp. 73-79 of Selected Problems 

in General Relativity, "Brandeis University 1960 Summer Insti- 
tute in Theoretical Physics Lecture Notes," distributed by 

W. A. Benjamin, Inc., New York. 
P. B. Moon, Developments in gamma-ray optics, Nature, 185, 427 

(1960). 
R. L. Mossbauer, Recoilless resonance absorption of gamma quanta 

in solids, Uspekhi Fiz. Nauk, 72, 658 (1960) (in Russian); Soviet 

Physics Uspekhi, 3, 866 (1961) (in English). 
R. V. Pound, On the weight of photons, Uspekhi Fiz. Nauk, 72, 673 

(1960) (in Russian); Soviet Physics Uspekhi, 3, 875 (1961) (in 

English). 
F. L. Shapiro, "Elementary Theory of the Mossbauer Effect," Press 

of Phys. Inst., Academy of Sciences, Moscow, 1960 (in Russian). 

F. L. Shapiro, The Mossbauer effect, Uspekhi Fiz. Nauk, 72, 685 
(1960) (in Russian); Soviet Physics Uspekhi, 3, 881 (1961) (in 
English); Fortschr. Physik, 9, 329 (1961) (in German). 

G. K. Wertheim, The Mossbauer effect: a tool for science, 
Nucleonics, 19, No. 1, 52 (1961). 

V. Weisskopf, The Mossbauer effect, in W. E. Brittin and B. W. 
Downs (eds.), "Lectures in Theoretical Physics," vol. 3, Inter- 
science, New York, 1961, pp. 70-80. 



7-2 RESEARCH PAPERS 

A. Abragam, Effect of ultrasonics on the emission and absorption 
of y radiation without recoil, Compt. rend., 250, 4334 (1960). 

A. Abragam and F. Boutron, Moment quadrupolaire du premier etat 
nucleaire excite du fer 57, Compt. rend., 252, 2404 (1961). 

C. Alff and G. K. Wertheim, Hyperfine structure of Fe 57 in yttrium - 
iron garnet from the Mossbauer effect, Phys. Rev., 122, 1414 
(1961). 

S. I. Aksenov, V. P. Alfimenkov, V. I. Lushchikov, Yu. M. Ostanevich, 
F. L. Shapiro, and Yen Wu-Kuang, Observation of resonance ab- 
sorption of gamma rays in Zn 67 , Zhur. Eksp. i Teoret. Fiz., 40, 
88 (1961) (in Russian); Soviet Physics JETP, 13, 62 (1961) (in 
English). 

I. Ya. Bar it, M. I. Podgoretzkii, and F. L. Shapiro, Several possible 
applications for the resonant scattering of y-rays, Zhur. Eksp. i 
Teoret. Fiz., 38, 301 (1960) (in Russian); Soviet Physics JETP, 11, 
218 (1960) (in English). 

R. Barloutaud, E. Cotton, J. L. Picou, and J. Quidort, Absorption 



88 THE MftSSBAUER EFFECT 

resonnante sans recul dy rayonnement y de 23.8 de Sn 119 , Compt. 
rend., 250, 319 (1960). 

R. Barloutaud, J. L. Picou, and C. Tzara, Diffusion resonnante du 
rayonnement y de 23.8 kev de Sn 119 emis sans recul, Compt. rend., 
250, 2705 (1960a). 

R. Bauminger, S. G. Cohen, A. Marinov, and S. Ofer, Hyperfine inter- 
actions in the ground state and first excited state of Dy 161 in dys- 
prosium iron garnet, Phys. Rev. Letters, 6, 467 (1961). 

R. Bauminger, S. G. Cohen, A. Marinov, and S. Ofer, Study of the in- 
ternal fields acting on iron-nuclei in iron garnet using the recoil 
free absorption in Fe 57 of the 14.4 kev gamma radiation from 
Fe 57m , Phys. Rev., 122, 743 (1961a). 

R. Bauminger, S. G. Cohen, A. Marinov, S. Ofer, and E. Segal, Study 
of the low temperature transitions in magnetite and the internal 
fields acting on iron nuclei in some spinel ferrites, using Moss- 
bauer absorption, Phys. Rev., 122, 1447 (1961b). 

P. J. Black and P. B. Moon, Resonant scattering of the 14 kev Fe 57 
y-ray and its interference with Rayleigh scattering, Nature, 188, 
481 (1960). 

A. J. F. Boyle, D. St. P. Bunbury, and C. Edwards, The isomer shift 
in Sn 119 and the quadrupole moment of the first excited state, 
Proc. Phys. Soc. (London) (to be published). 

A. J. F. Boyle, D. St. P. Bunbury, and C. Edwards, The nuclear 
Zeeman effect and quadrupole splitting in Sn 119 , Proc. Phys. Soc. 
(London), 77, 1062 (1961b). 

A. J. F. Boyle, D. St. P. Bunbury, and C. Edwards, Polarization of 
the conduction electrons in the ferromagnetic metals, Phys. Rev. 
Letters, 5, 553 (1960). 

A. J. F. Boyle, D. St. P. Bunbury, C. Edwards, and H. E. Hall, 
A chemical red shift of the recoilless y-emission of Sn 119m , Proc. 
Phys. Soc. (London), 76, 165 (1960a). 

A. J. F. Boyle, D. St. P. Bunbury, C. Edwards, and H. E. Hall, The 
Mossbauer effect in tin from 120 °K to the melting point, Proc. 
Phys. Soc. (London), 77, 129 (1961a). 

V. A. Bryukhanov, N. N. Delyagin, B. Zhvenglinskii, and V. S. 

Shpinel, The energy shifts of y -transitions observed in resonance 
absorption of gamma quanta in crystals, Zhur. Eksp. i Teoret. 
Fiz., 40, 713 (1961) (in Russian); Soviet Physics JETP, 13, 499 
(1961) (in English). 

A. Bussiere de Nercy, M. Langevin, and M. Spighel, Absorption 
resonnante du rayonnement y de Pholmium 166 et de Posmium 
193 sans recul de noyau, Compt. rend., 250, 1031 (1960). 

A. Bussiere de Nercy, M. Langevin, and M. Spighel, Absorption 

resonnante du rayonnement y sans recul du noyau de Ho 166 et Os 193 , 
J. phys. radium, 21, 288 (1960a). 

D. C. Champeney and P. B. Moon, Absence of Doppler shift for y-ray 



BIBLIOGRAPHY 89 

source and detector on same circular orbit, Proc. Phys. Soc. 
(London), 77, 350 (1961). 

G. Cocconi and E. E. Salpeter, Upper limit for the anisotropy of in- 
ertia from the Mossbauer effect, Phys. Rev. Letters, 4, 176 (1960). 

M. Cordey-Hayes, N. A. Dyson, and P. B. Moon, Width and intensity 
of the Mossbauer line in Fe 57 , Proc. Phys. Soc. (London), 75, 
810 (1960). 

P. P. Craig, J. G. Dash, A. D. McGuire, D. E. Nagle, and R. R. 
Reiswig, Nuclear resonance absorption of gamma rays in Ir 191 , 
Phys. Rev. Letters, 3, 221 (1959). 

P. P. Craig, D. E. Nagle, and D. R. F. Cochran, Zeeman effect in 
the recoilless y-ray resonance of Zn 67 , Phys. Rev. Letters, 4, 
561 (1960). 

P. P. Craig, D. E. Nagle, and R. D. Reiswig, Resonant absorption of 
gamma radiation in superconductors, Phys. and Chem. Solids, 17, 
168 (1960a). 

P.P. Craig, D. Nagle, and R. D. Taylor, Mossbauer effect in super- 
conducting indium, Nuovo cimento, 22, 402 (1961). 

T. E. Cranshaw, J. P. Schiffer, and A. B. Whitehead, Measurement 
of the gravitational red shift using the Mossbauer effect in Fe 57 , 
Phys. Rev. Letters, 4, 163 (1960). 

J. G. Dash, R. D. Taylor, P. P. Craig, D. E. Nagle, D. R. F. Cochran, 
and W. E. Keller, Mossbauer effect in Fe 57 at very low tempera- 
ture, Phys. Rev. Letters, 5, 152 (1960). 

J. G. Dash, R. D. Taylor, D.E. Nagle, P.P.Craig, and W. M. Visscher, 
Polarization of Co 57 in Fe metal, Phys. Rev., 122, 1116 (1961). 

S. DeBenedetti, G. Lang, and R. Ingalls, Electric quadrupole split- 
ting and the nuclear volume effect in the ions of Fe 57 , Phys. Rev. 
Letters, 6, 60 (1961). 

N. N. Delyagin, V. S. Shpinel, V. A. Bryukhanov, and B. Zhvenglinskii, 
Hyperfine structure of y-rays caused by quadrupole interaction in 
a crystal lattice, Zhru. Eksp. i Teoret. Fiz., 39, 220 (1960) (in 
Russian); Soviet Physics JETP, 12, 159 (1961) (in English). 

N. N. Delyagin, V. S. Shpinel, V. A. Bryukhanov, and B. Zhvenglinskii, 
Nuclear Zeeman-effect in Sn 119 , Zhur. Eksp. i Teoret. Fiz., 39, 
894 (1960a) (in Russian); Soviet Physics JETP, 12, 619 (1961) (in 
English). 

G. DePasquali, H. Frauenfelder, S. Margulies, and R. N. Peacock, 
Nuclear resonance absorption and nuclear Zeeman effect in Fe 57 , 
Phys. Rev. Letters, 4, 71 (1960). 

I. P. Dziub and A. F. Lubchenko, On the theory of the Mossbauer ef- 
fect, Doklady Akad. Nauk SSSR, 136, 66 (1961) (in Russian); Phys. 
Express, 3, No. 9, 34 (1961) (in English); Soviet Physics Doklady, 
6, 33 (1961) (in English). 

P. A. Flinn and S. L. Ruby, Local magnetic fields in Fe-Al alloys, 
Phys. Rev., 124, 34 (1961). 



90 THE MOSSBAUER EFFECT 

H. Frauenfelder, D. R. F. Cochran, D. E. Nagle, and R. D. Taylor, 

Internal conversion from resonance absorption, Nuovo cimento, 

19, 183 (1961). 
H. Frauenfelder, D. E. Nagle, R. D. Taylor, D. R. F. Cochran, and 

W. M. Visscher, Elliptical polarization of Fe 57 gamma rays (to 

be published). 
J. Gastebois and J. Quidort, Mise en evidence du moment quadru- 

polaire du noyau de fer 57, Compt. rend, 253, 1257 (1961). 

A. Gelberg, Winkelverteilung und zirkulare Polarisation der 
y-Strahlung im Mossbauereffekt, Inst, de Fiz. Atomica Maguerle- 
Bucuresti Rept. 16. 

L. Grodzins and F. Genovese, Experimental investigation of parity 
conservation in the 14.4 kev gamma transition of Fe 57 , Phys. Rev., 
121, 228 (1961). 

L. Grodzins and A. Phillips, Measurement of the refractive index of 
lucite by recoilless resonance absorption, Phys. Rev., 124, 774 
(1961a). 

M. N. Hack and M. Hamermesh, Effect of radiofrequency resonance 
on the natural line form, Nuovo cimento, 19, 546 (1961). 

R. V. Hanks, Pressure dependence of the Mossbauer effect, Phys. 
Rev., 124, 1319 (1961). 

S. S. Hanna, J. Heberle, C. Littlejohn, G. J. Perlow, R. S. Preston, 
and D. H. Vincent, Observation on the Mossbauer effect in Fe 57 , 
Phys. Rev. Letters, 4, 28 (1960). 

S. S. Hanna, J. Heberle, C. Littlejohn, G. J. Perlow, R. S. Preston, 
and D. H. Vincent, Polarized spectra and hyperfine structure in 
Fe 57 , Phys. Rev. Letters, 4, 177 (1960a). 

S. S. Hanna, J. Heberle, G. J. Perlow, R. S. Preston, and D. H. Vin- 
cent, Direction of the effective magnetic field at the nucleus in 
ferromagnetic iron, Phys. Rev. Letters, 4, 513 (1960b). 

S. S. Hanna, L. Meyer-Schutzmeister, R. S. Preston, and D. H. Vin- 
cent, Nuclear Zeeman effect in Sn 119 , Phys. Rev., 120, 2211 
(1960c). 

S. M. Harris, Quantum mechanical calculation of Mossbauer trans- 
mission, Phys. Rev., 124, 1178 (1961). 

H. J. Hay, J. P. Schiffer, T. E. Cranshaw, and P. A. Egelstaff, 
Measurement of the red shift in an accelerated system using the 
Mossbauer effect in Fe 57 , Phys. Rev. Letters, 4, 165 (1960). 

B. Hoffman, Noon-midnight red shift, Phys. Rev., 121, 337 (1960). 
R. E. Holland, F. J. Lynch, G. J. Perlow, and S. S. Hanna, Time 

spectra of filtered resonance radiation of Fe 57 , Phys. Rev. Let- 
ters, 4, 181 (1960). 
S. Jha, R. K. Gupta, H. G. Devare, G. C. Pramila, and R. Srinivasa 
Raghavan, Recoilless emission and absorption of 26 kev gamma- 
ray of Dy 161 , Nuovo cimento, 19, 682 (1961). 

C. E. Johnson, M. S. Ridout, T. E. Cranshaw, and P. E. Madsen, 



BIBLIOGRAPHY 91 

Hyperfine field and atomic moment of Fe in ferromagnetic alloys, 

Phys. Rev. Letters, 6, 450 (1961). 
B. D. Josephson, Temperature dependent shift of y-rays emitted by 

a solid, Phys. Rev. Letters, 4, 341 (1960). 
Yu. Kagan, Anisotropy of the Mossbauer effect, Doklady Akad. Nauk 

SSSR, 140, 794 (1961). 
Yu. Kagan, Determination of the frequency spectrum of phonons in 

crystals, Zhur. Eksp. i Teoret. Fiz., 40, 312 (1961) (in Russian); 

Soviet Physics JETP, 13, 211 (1961a) (in English). 
M. Kalvius, P. Kienle, K. Bockmann, and H. Eicher, Hyperfein- 

strukturaufspal$ung von riickstossfreien y-Linien: II. Das 8,42 

kev Niveau in Tm 169 , Z. Physik, 163, 87 (1961). 
E. Kankeleit, Untersuchung von Konversions-Elektronen beim 

Mossbauereffekt am Wolfram 182 mit einem magnetischen 

Spektrometer, Z. Physik, 164, 442 (1961). 
A. Kastler, Sur la possibilite de mettre en evidence la coherence 

de phase dans la diffusion de resonance des rayons y par des 

noyaux atomiques, Compt. rend., 250, 509 (1960). 
M. V. Kazarnovskii, Theory of resonant interaction of y-ray with 

crystals, Zhur. Eksp. i Teoret. Fiz., 38, 1652 (I960) (in Russian); 

Soviet Physics JETP, 11, 1191 (1960) (in English). 
W. H. Kelly, V. J. Folen, M. Hass, W. N. Schreiner, and G. G. 

Beard, Magnetic field at the nucleus in spinel-type crystals, 

Phys. Rev., 124, 80 (1961). 
O. C. Kistner and A. W. Sunyar, Evidence for quadrupole interaction 

of Fe 57m and influence of chemical binding on nuclear gamma-ray 

energy, Phys. Rev. Letters, 4, 412 (1960). 

0. C. Kistner, A. W. Sunyar, and J. B. Swan, Hyperfine structure of 
the 24 kev transitions in Sn 119 , Phys. Rev. 123, 179 (1961). 

1. Y. Krause and G. Liiders, Experimentelle Priifung der Relativitats- 

theorie mit Kernresonanzabsorption, Naturwissenschaften, 48, 34 
(1961). 

M. A. Krivoglaz, Effect of diffusion on the scattering of neutrons and 
photons by crystal imperfections, and on the Mossbauer effect, 
Zhur. Eksp. i Teoret. Fiz., 40, 1812 (1961) (in Russian); Soviet 
Physics JETP, 13, 1273 (1961) (in English). 

W. E. Lamb, Jr., Possible use of highly monochromatic gamma rays 
for microwave spectroscopy, in C. H. Townes (ed.), "Quantum 
Electronics, a Symposium," Columbia University Press, New 
York, 1960, p. 588. 

L. L. Lee, L. Meyer-Schiitzmeister, J. P. Schiffer, and D. Vincent, 
Nuclear resonance absorption of gamma rays at low temperatures, 
Phys. Rev. Letters, 3, 223 (1959). 

H. J. Lipkin, The Debye -Waller factor in Mossbauer interference ex- 
periments, Phys. Rev., 123, 62 (1961). 



92 THE MOSSBAUER EFFECT 

H.J. Lipkin, Some simple features of the Mossbauer effect, Ann. 
Phys., 9, 332 (1960). 

H. J. Lipkin, Some simple features of the Mossbauer effect, II, Ann. 
Phys. (in press). 

W. Low, The effect of relaxation phenomena on Mossbauer experi- 
ments, in G.. M. Graham and A. C. Hollis-Hallett (eds.), "Proceed- 
ings of the Vllth International Conference on Low Temperature 
Physics, " University of Toronto Press, Toronto, 1961, pp. 20-22. 

F. J. Lynch, R. E. Holland, and M. Hamermesh, Time dependence of 
resonantly filtered gamma rays from Fe 57 , Phys. Rev., 120, 513 
(1960). 

V. A. Lyubimov and A. I. Alikhanov, Influence of magnetic fields on 
resonance absorption of gamma rays, Zhur. Eksp. i Teoret. Fiz., 

38, 1912 (1960) (in Russian); Soviet Physics JETP, 11, 1375 (1960) 
(in English); Izvest. Akad. Nauk SSSR, 24, 1076 (1960) (in Russian); 
Bull. acad. sci. URSS, 24, 1084 (in English). 

K. J. Major, Recoil-free resonant and nonresonant scattering from 
Fe 57 (to be published). 

S. V. Maleev, On the use of the Mossbauer effect for studying lo- 
calized oscillations of atoms in solids, Zhur. Eksp. i Teoret. Fiz., 

39, 891 (1960) (in Russian); Soviet Physics JETP, 12, 617 (1961) 
(in English). 

S. Margulies, The Mossbauer effect in iron-57 (unpublished thesis, 

Univ. Illinois, 1962) [Tech. Rept. 28, on Nonr 1834(05), Physics 

Dept., Univ. Illinois (October 1961)]. 
S. Margulies and J. R. Ehrman, Transmission and line broadening of 

resonance radiation incident on a resonance absorber, Nuclear 

Instr., 12, 131 (1961a). 
Luise Meyer -Schiitzmeister, R. S. Preston, and S. S. Hanna, Internal 

magnetic fields in manganese-tin alloys, Phys. Rev., 122, 1717 

(1961). 
K. P. Mitrofanov and V. S. Shpinel, Resonance absorption of the 

23.8 kev y-ray of Sn 119 observed by means of internally converted 

electrons, Zhur. Eksp. i Teoret. Fiz., 40, 983 (1961) (in Russian); 

Soviet Physics JETP, 13, 686 (1961) (in English). 
P. B. Moon, Interference between Rayleigh and nuclear resonant 

scattering of gamma rays, Proc. Royal Soc. (London), A263, 309 

(1961). 
M. Morita, Possible parity and time -reversal experiments using 

the Mossbauer effect, Phys. Rev., 122, 1525 (1961). 
R. L. Mossbauer, Kernresonanzfluoreszenz von Gammastrahlung in 

Ir 191 , Z. Physik, 151, 124 (1958). 
R. L. Mossbauer, Kernresonanzabsorption von Gammastrahlung in 

Ir 191 , Naturwissenschaften, 45, 538 (1958a). 
R. L. Mossbauer, Kernresonanzabsorption von Gammastrahlung in 

Ir 191 , Z. Naturforsch., 14a, 211 (1959). 



BIBLIOGRAPHY 93 

R. L. Mossbauer, F. W. Stanek, and W. H. Wiedemann, Hyperfein- 
strukturaufspaltung von ruckstossfreien y-Linien: I. Das 
80,6 kev-Niveau in Er 166 , Z. Physik, 161, 388 (1961). 

R. L. Mossbauer and W. H. Wiedemann, Kernresonanz absorption 
nicht Dopplerverbreiterter Gammastrahlung in Re 187 , Z. Physik, 
159, 33 (1960). 

D. E. Nagle, P. P. Craig, P. Barrett, R. D. Taylor, and D. R. F. 
Cochran, Internal fields at low temperatures in CoPd alloys, 
Phys. Rev. (to be published). 

D. Nagle, P.P. Craig, J. G. Dash, and R. R. Reiswig, Nuclear reso- 
nance fluorescence in Au 197 , Phys. Rev. Letters, 4, 237 (1960). 

D. E. Nagle, P.P. Craig, and W. E. Keller, Ultra high resolution 
y-ray resonance in Zn 67 , Nature, 186, 707 (1960a). 

D. E. Nagle, H. Frauenf elder, R. D. Taylor, D. R. F. Cochran, and 
B. T. Matthias, Temperature dependence of the internal field in 
ferromagnets, Phys. Rev. Letters, 5, 364 (1960b). 

F. E. Obenshain and H. H. F. Wegener, Mossbauer effect with Ni 61 , 
Phys. Rev., 121, 1344 (1961). 

S. Ofer, P. Avivi, R. Bauminger, J. Marinov, and S. G. Cohen, Nu- 
clear resonance absorption in Dy 161 situated in Dy 2 3 and dyspro- 
sium iron garnet, Phys. Rev., 120, 406 (1960). 

G. J. Perlow, S. S. Hanna, M. Hamermesh, C. Littlejohn, D. H. 
Vincent, R. S. Preston, and J. Heberle, Polarization of nuclear 
resonance radiation in ferromagnetic Fe 57 , Phys. Rev. Letters, 
4, 74 (1960). 

J. Petzold, Einige Bermerkungen zur Theorie des Mossbauer- 

Effektes, Z. Physik, 163, 71 (1961). 
J. Petzold, "Theorie des M6ssbauer-Effektes, ,, Sitzber. Heidelberg 

Akad. der Wissen. (Springer Verlag, Heidelberg, 1961a). 
J. L Picou, J. Quidort, R. Barloutaud, and E. Cotton, Absorption 

resonnante des y sans recul dans la des integration du premier etat 

excite de Sn 119 (to be published). 
M. I. Podgoretskii and I. I. Roizen, Radiation of a nucleus in the 

presence of unexcited nuclei of the same type, Zhur. Eksp. i 

Teoret. Fiz., 39, 1473 (1960) (in Russian); Soviet Physics JETP 

12, 1023 (1960) (in English); Physics Express, 3, No. 7, 22 (1961) 

(in English). 
M. I. Podgoretskii and A. V. Stepanov, The Doppler width of emission 

and absorption lines, Zhur. Eksp. i Teoret. Fiz., 40, 561 (1961) 

(in Russian); Soviet Physics JETP, 13, 393 (1961) (in English). 
R. V. Pound, G. B. Benedek, and R. Drever, Effect of hydrostatic 

compression on the energy of the 14.4 kev gamma ray from Fe 57 

in iron, Phys. Rev. Letters, 7, 405 (1961). 
R. V. Pound and G. A. Rebka, Jr., Apparent weight of photons, Phys. 

Rev. Letters, 4, 337 (1960). 
R. V. Pound and G. A. Rebka, Jr., Attempts to detect resonance scat- 



94 THE MOSSBAUER EFFECT 

tering in Zn 67 ; the effect of zero point vibrations, Phys. Rev. Let- 
ters, 4, 397 (1960a). 
R. V. Pound and G. A. Rebka, Jr., Gravitational red- shift in nuclear 

resonance, Phys. Rev. Letters, 3, 439 (1959). 
R. V. Pound and G. A. Rebka, Jr., Resonant absorption of the 14.4 kev 

y-ray from 0.10 jiisec Fe 57 , Phys. Rev. Letters, 3, 554 (1959a). 
R. V. Pound and G. A. Rebka, Jr., Variation with temperature of the 

energy of recoil-free y-rays from solids, Phys. Rev. Letters, 4, 

274 (1960). 
R. S. Raghavan, On the possibility of a new phase determining method 

applying the Mossbauer effect (to be published). 
S. L. Ruby and D. I. Bolef, Acoustically modulated y-rays from Fe 57 , 

Phys. Rev. Letters, 5, 5 (1960). 
S. L. Ruby, L. M. Epstein, and K. H. Sun, Mossbauer effect in ferro- 

cyanide, Rev. Sci. Instr., 31, 580 (1960). 
S. L. Ruby and G. Shirane, Magnetic anomaly in FeTi0 3 -a! Fe 2 O s sys- 
tem by Mossbauer effect, Phys. Rev., 123, 1239 (1961). 
M. Ruderfer, First order terrestrial ether drift experiment using 

the Mossbauer radiation, Phys. Rev. Letters, 5, 191 (1960); 

correction: Phys. Rev. Letters, 7, 361 (1961). 
J. P. Schiffer and W. Marshall, Recoilless resonance absorption of 

gamma rays in Fe 57 , Phys. Rev. Letters, 3, 556 (1959). 
C. W. Sherwin, H. Frauenfelder, E. L. Garwin, E. Luscher, S. 

Margulies, and R. N. Peacock, A search for the anisotropy of 

inertia using the Mossbauer effect in Fe 57 , Phys. Rev. Letters, 

4, 399 (1960). 

C. W. Sherwin, Some recent experimental tests of the "clock para- 
dox," Phys. Rev., 120, 17 (1960a). 

G. Shirane, D. E. Cox, and S. L. Ruby, Mossbauer study of isomer 
shift, quadrupole interaction, and hyperfine field in several oxides 
containing Fe 57 , Phys. Rev. (to be published). 

D. A. Shirley, Interpretation of the isomeric chemical shifts in Au 197 , 
Phys. Rev., 124, 354 (1961). 

D. A. Shirley, M. Kaplan, and P. Axel, Recoil-free resonant absorp- 
tion in Au 197 , Phys. Rev., 123, 816 (1961a). 

V. H. Shpinel, V. A. Bryukhanov, and N. N. Delyagin, Effect of tem- 
perature on the hyperfine structure of gamma radiation, Zhur. 
Eksp. i Teoret. Fiz., 40, 1525 (1961) (in Russian). 

K. S. Singwi and A. Sjolander, Resonance absorption of nuclear y-rays 
and the dynamics of atomic motions, Phys. Rev., 120, 2211 (1960). 

V. V. Sklyarevskii, B. N. Samoilov, and E. P. Stepanov, Temperature 
dependence of the hyperfine splitting of dysprosium levels in para- 
magnetic dysprosium oxide, Zhur. Eksp. i Teoret. Fiz., 40, 1875 
(1961) (in Russian). 

H. S. Snyder and G. C. Wick, Broadening of the Mossbauer line, Phys. 
Rev., 120, 128 (1960). 



BIBLIOGRAPHY 95 

I. Solomon, Effet Mossbauer dans la pyrite et la marcassite, Compt. 

rend., 250, 3828 (1960). 
I. Solomon, Mesure par effet Mossbauer de champs locaux dans 

divers composes du fer, Compt. rend., 251, 2675 (1960). 
R. D. Taylor, A low temperature thermometer utilizing the Mossbauer 

effect, in C. M. Herzfeld (ed.), ''Temperature, Its Measurement 

and Control in Science and Industry," Reinhold, New York, 1962. 
G. T. Trammel, Elastic scattering at resonance from bound nuclei 

(to be published). 
P.-K. Tseng, N. Shikazono, H. Takehoshi, and T. Shoji, Temperature 

dependence of nuclear resonance absorption line width in Dy 161 , 

J. Phys. Soc. Japan, 16, 1790 (1961). 
C. Tzara, Diffusion des photons sur les atomes et les noyaux dans 

les cristaux, J. phys. radium, 22, 303 (1961). 
C. Tzara, Sur l'excitation resonnante deniveaux nucleaires metastables 

de vie tres longue, Compt. rend., 250, 1466 (1960). 
C. Tzara and R. Barloutaud, Recoilless Rayleigh scattering in solids, 

Phys. Rev. Letters, 4, 405 (1960a). 
W. M. Visscher, Neutrino detection by resonance absorption in 

crystals at low temperature, Phys. Rev., 116, 1581 (1959). 
W. M. Visscher, Recoilless emission by an impurity atom (to be 

published). 
W. M. Visscher, Study of lattice vibrations by resonance absorption 

of nuclear gamma rays, Ann. Phys., 9, 194 (1960). 
J. van Kranendonk, Theoretical aspects of the Mossbauer effect, in 

G. M. Graham and A. C. Hollis-Hallett (eds.), "Proceedings of 

the Vnth International Conference on Low Temperature Physics," 

University of Toronto Press, Toronto, 1961, pp. 9-20. 

F. E. Wagner, F. W. Stanek, P. Kienle, and H. Eicher, Hyperfein- 
strukturauf spaltung von riickstossfreien y -Linien: in. Das 
84 kev-Niveau in Yb 170 , Z. Physik (to be published). 

L. R. Walker, G. K. Wertheim, and V. Jaccarino, Interpretation of 
the Fe 57 isomer shift, Phys. Rev. Letters, 6, 98 (1961). 

H. H. F. Wegener and F. E. Obenshain, Mossbauer effect for Ni 61 
with applied magnetic fields, Z. Physik, 163, 17 (1961). 

G. K. Wertheim, Hyperfine structure of divalent and trivalent Fe 57 in 
cobalt oxide, Phys. Rev., 124, 764 (1961). 

G. K. Wertheim, Hyperfine structure of Fe 57 in paramagnetic and 

antiferromagnetic FeF 2 from the Mossbauer effect, Phys. Rev., 

121, 63 (1961a). 
G. K. Wertheim, Measurement of local fields at impurity Fe 57 atoms 

using the Mossbauer effect, Phys. Rev. Letters, 4, 403 (1960). 
G. K. Wertheim, Mossbauer effect: applications to magnetism, 

J. Appl. Phys. (Suppl), 32, 110S (1961b). 
G. K. Wertheim and J. H. Wernick, Fe 57 Mossbauer effect on Cu-Ni 

alloys, Phys. Rev., 123, 755 (1961c). 



96 THE MOSSBAUER EFFECT 

W. H. Wiedemann, P. Kienle, and F. Pobell, Mossbauereffekt im 
normalleitenden und supraleitenden Zustand von Sn 119 , Z. Physik, 
165 (1961). 

C. S. Wu, Y. K. Lee, N. Benczer-Koller, and P. Simms, Frequency 
distribution of resonance line versus delay time, Phys. Rev. 
Letters, 5, 432 (1960). 

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merieverschiebung der 14.4 kev y-Linie von Fe 57 in metall- 
organischen Eisenverbindungen, Z. Physik (in press). 

L. G. Zastavenko and M. I. Podgoretskii, The effect of external fields 
on the angular correlations and resonance processes during quan- 
tum transitions, Zhur. Eksp. i Teoret. Fiz., 39, 1023 (1960) (in 
Russian); Soviet Physics JETP, 12, 711 (1961) (in English). Ex- 
cerpts also translated in Phys. Express, 3, No. 5, 21 (1961). 

7-3 CONFERENCE REPORTS 

"Mossbauer Effect; Recoilless Emission and Absorption of Gamma 
Rays," Hans Frauenf elder and Harry Lustig (eds.) (Discussions 
of On-Site Conference of the Advisory Committee to the Directo- 
rate of Solid State Sciences, Air Force Office of Scientific Re- 
search, Univ. Illinois, Allerton House, June 6-7, 1960). Available 
only as AF-TN 60-698. This unpublished report is known in the 
field as "The First Mossbauer Conference Report." 

Proceedings of the Second Mossbauer Conference, Paris, Sept. 13-16, 
1961, A. Schoen and D. M. J. Compton (eds.), Wiley, New York, 
tentative publication Spring 1962. 



REPRINTS AND 
TRANSLATIONS 



99 



The Effect of Collisions upon the Doppler Width of Spectral Lines 

R. H. DtcKE 

Palmer Physical Laboratory, Princeton University, Princeton, New Jersey 
(Received September 17, 1952) 



Quantum mechanically the Doppler effect results from the recoil momentum changing the translational 
energy of the radiating atom. The assumption that the recoil momentum is given to the radiating atom is 
shown to be incorrect if collisions are taking place. If the collisions do not cause broadening by affecting the 
internal state of the radiator, they result in a substantial narrowing of the Doppler broadened line. 



QUANTUM mechanically, the Doppler effect re- 
sults from the recoil momentum given to the 
radiating system by the emitted photon. 1 This recoil 
momentum implies a change in the kinetic energy of the 
radiating atom which is in turn mirrored by a corre- 
sponding change in the photon's energy. This change 
in the photon's energy is proportional to the component 
of the atom's velocity in the direction of emission of the 
photon and leads to the normal expression for the 
Doppler effect. Since for gas pressures commonly en- 
countered the fraction of the time that an atom is in 
collision is negligibly small, it might seem reasonable to 
assume that the recoil momentum is absorbed by the 
single radiating atom or molecule rather than by an 
atomic aggregate. In this case the Doppler breadth 



a=-£ x 



a-* 



a=f 



V (|-£) 



EMITTED , 
v ('|-£) FREQUENCY 



Fig. 1. Spectral distribution of radiation emitted by an atom 
confined to a one-dimensional box of width a. 



E. Fermi, Revs. Modern Phys. 4, 105 (1932). 



would, within limits, be pressure-independent. Actu- 
ally, under certain circumstances, this assumption is 
far from correct. Collisions which do not affect the in- 
ternal state of the radiating system have a large effect 
upon the Doppler breadth. 

The effect of collisions upon the Doppler effect is 
best illustrated with a simple example treated first 
classically and then quantum mechanically. Assume 
that the radiating atom, but not the radiation, is con- 
fined to a one-dimensional well of width a, and that it 
moves back and forth between the two walls with a 
speed v. The wave emitted by the atom is frequency 
modulated with the various harmonics of the oscillation 
frequency of the atom in the square well. For negligible 
collision and radiation damping, the spectral distribu- 
tion of the emitted radiation is obtained from a Fourier 
series. A set of equally spaced sharp lines is obtained. 
They occur at the non-Doppler shifted frequency plus- 
or minus-integral multiples of the oscillation frequency 
of the atom in the square well. The intensity distribu- 
tion of these lines is shown for several values of a/X in 
Fig. 1. 

In the quantum-mechanical description of this ex- 
ample, the radiating system possesses two types of 
energy, internal and external. The external energy is 
the quantized energy of the atomic center-of-mass 
moving in the one-dimensional square well. In a transi- 
tion in which a photon is absorbed or emitted, both the 
internal and external quantum numbers may change. 
The frequency of the emitted photon is 

^ m =H-(V8Ma 2 )(n 2 -m 2 ). 

Here v is the frequency of the non-Doppler shifted 
line, M is the mass of the radiator, and n, m are in- 
tegers. A calculation of the transition probabilities 
gives results for the intensities which are for large n 
and m essentially the same as the classical results 
(Fig. 1). 

The introduction of a Maxwellian distribution in v 
in the case of the classical calculation leads to a con- 
tinuous distribution very similar to a normal Doppler 
distribution plus a sharp non-Doppler broadened line 
(see Fig. 2). The fraction of the energy radiated in the 
sharp line is 

sin 2 (7ra/X) 



(wa/XY 



472 



100 



473 



DOPPLER WIDTH OF SPECTRAL LINES 



The sharp line has its origin in the fact that, for a non- 
integral value of 7ra/X, the normal unshifted frequency 
is emitted by all atoms independent of their speed. 
Since for a>^\ the dominant noncentral lines in 
Fig. 1 are always close to the normal Doppler shifted 
frequencies, the broad distribution has a line contour 
nearly identical with the normal Doppler line. For 
a<|X, the distribution increases in breadth but be- 
comes much weaker. 

For the quantum-mechanical treatment, a Maxwell- 
Boltzmann distribution among the various energy 
levels leads to a fine complex of lines having fre- 
quencies v nm . If the zero-point energy of oscillation of 
the atom in the well is very small compared with kT, 
the degenerate frequency v= v nn is usually the most 
intense single frequency emitted. For a small amount 
of collision or natural broadening, the complex of lines 
becomes a continuous distribution (Fig. 2) essentially 
identical with that given by the classical calculation. 
Note that although the atom is in contact with the 
walls of the cavity only an infinitesimal part of the 
time, the probability of the photon's momentum being 
given to trTe~walls rather than to the atom is finite, being 

sin 2 (7ra/X) 



(Tra/X) 2 



For a gas confined to a large volume but with a 
mean free path small compared with a wavelength, the 
shape of a Doppler broadened line has been calculated 
treating the radiation classically and using a statistical 
procedure. In this treatment the phase of the radiation 
emitted as a function of the time is given by the posi- 
tion of the radiator as a function of the time. The 
probability distribution of position given by diffusion 
theory is used to calculate the mean intensity as a 
function of frequency. Substantially the same result is 
obtained also quantum mechanically, using a method 
similar to Foley's. 2 This quantum-mechanical calcula- 
tion is valid only if the recoil energy of the radiator is 
small compared with kT. Assuming that the Doppler 

1 H. M. Foley, Phys. Rev. 69, 616 (1946). 




"V FREQUENCY 

Fig. 2. Doppler broadened line of a gas in a one-dimensional box. 

effect is the only appreciable source of the line breadth, 
it is found that the line has a Lorentz rather than 
Gaussian shape. The line contour is given by 



/(«) = /«r 



2irD/\ 2 



(a- V y+(2irD/\*y 



The width of the line at half-intensity is, in cycles per 
second, 4xZ)/X 2 . Here D is the self-diffusion constant of 
the gas. This line width is roughly 2.8Z./X times that 
of a normal Doppler broadened line (L is the mean free 
path). Therefore, under those conditions for which the 
calculation is valid, the line breadth which is wholly 
Doppler is greatly reduced. 

Because of the requirement that the gas collisions 
should not influence the internal state of the radiator, 
the above results are ordinarily valid only for certain 
magnetic dipole transitions. Nuclear magnetic resonance 
absorption, paramagnetic resonance absorption, and 
5-state hyperfine transitions are examples of transitions 
which are but weakly affected by collisions. 



101 

^^^4*******4-*********«9-**-9-******4'********-9- -5- 4-4- -5- * 4- -3- -5- * * * * -5- * * 



Aus dem Institut fiir Physik im Max-Planck-Institut fiir medizinische Forschung, 

Heidelberg 

Kernresonanzfluoreszenz von Gammastrahlung in Ir 191 

Von 

Rudolf L. Mossbauer* 

Mit 8 Figuren im Text 
(Eingegangen am 9. Januar 1958) 

Die Kernresonanzabsorption der dem Zerfall von Os 191 folgenden 129 keV- Gamma- 
strahlung in Ir 191 wird untersucht. Der Wirkungsquerschnitt fiir die Resonanz- 
absorption wird als Funktion der Temperaturen von Quelle und Absorber im 
Temperaturbereich 90° K< T< 370° K gemessen. Die Lebenszeit r des 129 keV- 
Niveaus in Ir 191 ergibt sich zu (3,6+ J|g) 1CT 10 sec. Der Absorptionsquerschnitt 
zeigt bei tiefen Temperaturen einen starken Anstieg als Folge der Kristallbindung 
der Absorber- und Praparatsubstanzen. Die Theorie von Lamb uber die Resonanz- 
absorption langsamer Neutronen in Kristallen wird auf die Kernresonanzabsorption 
von Gammastrahlung ubertragen. Bei tiefen Temperaturen ergibt sich eine 
starke Abhangigkeit des Wirkungsquerschnittes fiir die Kernabsorption von der 
Frequenzverteilung im Schwingungsspektrum des Festkorpers. 

1. Einleitung 

Die Kernresonanzfluoreszenz von Gammastrahlung ist das kern- 
physikalische Analogon zu der bekannten Fluoreszenzerscheinung der 
Atomhulle : Anregung eines Kernniveaus durch Einstrahlung der eigenen 
Linie, wobei die Emission und Absorption in Kernen gleicher Art statt- 
findet. Die Quanten erfahren bei ihrer Emission bzw. Absorption 
Energieverluste infolge Abgabe von RuckstoBenergie an die emittieren- 
den bzw. absorbierenden Kerne, was zu einer Verschiebung der Emis- 
sionslinie gegeniiber der Absorptionslinie fuhrt. Bei Kernubergangen 
ist, umgekehrt wie bei optischen Obergangen, die durch den RiickstoB- 
energieverlust der Quanten bedingte Linienverschiebung immer groB 
gegen die naturliche Linienbreite, d.h. die Resonanzbedingung ist ver- 
letzt. Da jedoch die tatsachliche Breite der Linien durch die Tempera- 
turbewegung der Kerne in Quelle und Absorber bestimmt wird**, die 
zu Doppler-Verschiebungen der Quantenenergien fuhrt, wird fiir einen 
Teil der Quanten der RuckstoBenergieverlust kompensiert und die durch 
den RiickstoBeffekt verletzte Resonanzbedingung wiederhergestellt. 

* Neue Anschrift: Labor fiir technische Physik, Technische Hochschule 
Munchen. 

** Die naturliche Linienbreite kann in alien hier interessierenden Fallen gegen- 
iiber der Doppler-Breite vernachlassigt werden. 



102 

Kernresonanzfluoreszenz von Gammastrahlung in Ir 191 125 

Die hohen RiickstoBenergieverluste der Quanten bei Kerniiber- 
gangen ergeben bei Zimmertemperatur haufig nur eine geringe tJber- 
deckung der Emissions- und Absorptionslinien, d.h. die Resonanz- 
bedingung ist nur fur wenige Quanten erfiillt und der Wirkungsquer- 
schnitt fur den Resonanzeffekt wird unmeBbar klein. Wie zuerst Malm- 
fors [2] zeigte, laBt sich in giinstigen Fallen durch Temperaturerhohung 
ein meBbarer Resonanzfluoreszenzeffekt erzielen*. Diese Methode der 
Temperaturerhohung, durch thermische Verbreiterung der Linien eine 
starkere Uberlagerung der Emissions- und Absorptionslinien zu er- 
reichen, wurde seitdem in einer Reihe von Arbeiten erfolgreich ange- 
wandt [3]. Diese Experimente erfolgten in Form von Streuversuchen, 
wobei jeweils die an den Kernen resonanzgestreuten Quanten von dem 
Untergrund der elastisch gestreuten und der durch den Compton-Effekt 
gestreuten Strahlung abgetrennt werden muBten. Die Messungen 
muBten im allgemeinen auf Quellen beschrankt werden, die keine 
hartere Gammastrahlung emittierten als die untersuchte Resonanz- 
strahlung, um im Nachweiskristall eine Uberdeckung des Photomaxi- 
mums der resonanzgestreuten Quanten durch das Compton-Kontinuum 
harterer Linien zu verhindern. Die Bestimmung des Kernfluoreszenz- 
effektes durch Messung der resonanzgestreuten Strahlung bietet bei 
weicher Gammastrahlung zwei zusatzliche Schwierigkeiten : 

1 . Mit abnehmender Energie wird es immer schwieriger, die Compton- 
Streustrahlung von der elastischen Streustrahlung zu unterscheiden, 
wegen des abnehmenden Energieunterschiedes zwischen primaren und 
Compton-gestreuten Quanten. Erschwerend wirkt sich aus, daB das 
Auflosungsvermogen der Szintillationsspektrometer mit abnehmender 
Energie abnimmt. AuBerdem steigt der Wirkungsquerschnitt fiir die 
Rayleigh-Streuung bei niedrigen Energien stark an [4]. 

2. Der Wirkungsquerschnitt fiir die Resonanzfluoreszenz ist umge- 
kehrt proportional zur Lebenszeit x y desResonanzniveaus**. Fur Lebens- 
zeitmessungen eignen sich daher gerade die kurzlebigen magnetischen 
Dipol (Ml)- und elektrischen Quadrupol (£2)-t)bergange (r y <10 _10 sec), 
die mit der Methode der verzogerten Koinzidenzen nicht mehr erfaBt 
werden konnen. Die Strahlungsiibergange niedriger Multipolordnung 
zeigen bei niedrigen Energien eine betrachtliche Kon version. Bei 
der Resonanzfluoreszenz wird von den resonanzabsorbierten Quanten 
nur der Bruchteil (1+a) -1 (a = Konversionskoeffizient) wieder als 
Quant reemittiert und nur dieser Bruchteil der resonanzabsorbierten 
Quanten steht fiir die Messung zur Verfiigung. 

* Einen t)berblick iiber die verschiedenen Verfahren zur Kompensation der 
RiickstoBenergieverluste gibt Malmfors [2]. 

** Diese Beziehung gilt nicht mehr fiir Festkorper bei tiefen Temperaturen ; 
vgl. Abschnitt 3. 



103 

126 Rudolf L. Mossbauer: 

Die genannten Schwierigkeiten lassen sich umgehen, wenn der Kern- 
resonanzeffekt in Absorption gemessen wird. Da jedoch der Effekt, 
besonders bei weicher Gammastrahlung, sehr klein ist gegenliber den 
Absorptionseffekten der Atomhulle, werden bei einem Absorptions- 
experiment zur Messung der Lebenszeit eines Kernniveaus extreme An- 
forderungen an die Genauigkeit und Stabilitat der MeBapparaturen 
gestellt. Dafiir bietet ein Absorptionsexperiment gegeniiber einem 
Streustrahlungsversuch den Vorteil eines urn einen Faktor 1 + a hoheren 
Wirkungsquerschnittes und ermoglicht eine einfachere Interpretation 
der MeBergebnisse. 

Die Kernresonanzfluoreszenz besitzt besonderes Interesse im Energie- 
gebiet weicher Gammastrahlung, da bei tiefen Temperaturen in diesem 
Energiegebiet bei Festkorpern Einflusse der chemischen Bindung zu 
erwarten sind. 

In der vorliegenden Arbeit wurde durch ein Gamma-Absorptions- 
experiment die Lebenszeit r y des 129 keV-Niveaus in Ir 191 bestimmt. 
Untersuchungen bei der Temperatur des fliissigen 2 ergaben einen 
starken EinfluB der chemischen Bindung auf den Wirkungsquerschnitt 
fur die Kernabsorption. Der aufgefundene Bindungseffekt wurde mit 
Hilfe der Theorie von Lamb, die zu diesem Zweck fur den Fall der 
Emissionslinie erweitert wurde, theoretisch gedeutet. Der Bindungs- 
effekt ist sehr empfindlich vom Schwingungsspektrum des Festkorpers 
abhangig. 

2. Kernresonanzfluoreszenz und chemische Bindung 
Die Absorptions- und Emissionslinien sind ihrer Lage und ihrer Form 
nach vom Bindungszustand abhangig. Insbesondere ist der RuckstoB- 
energieverlust der emittierten und auch der resonanzabsorbierten Quan- 
ten abhangig von den Energieaufnahme- und Energieabgabemoglich- 
keiten der Systeme, denen die betrachteten Kerne angehoren (z.B. 
Molekule oder Kristalle). 

Ein freier Kern der Masse m ubernimmt bei Emission eines Quants 
der Energie E eine RiickstoBenergie R, die gegeben ist durch 

R = E 2 j2mcK (1) 

Im Falle einer chemischen Bindung des Kernes in einem Kristall muB 
der Kristall die RiickstoBenergie als innere Energie aufnehmen. Wegen 
der Quantelung der inneren Energie konnen jedoch beim RiickstoB nur 
diskrete Energien aufgenommen werden und die RiickstoBenergie 
hangt ab von den Wahrscheinlichkeiten flir die Anregung der Gitter- 
schwingungen des Kristalles. Bei Temperaturen T, die groB sind gegen 
die Debyesche Temperatur des Kristalles, ist die statistische Ge- 
schwindigkeitsverteilung der Kerne unabhangig von der Bindung und 



104 

Kernresonanzfluoreszenz von Gammastrahlung in Ir 191 127 

es erfolgt eine ungehinderte Ubertragung der vollen RiickstoBenergie 
nach (1). Mit abnehmender Temper atur gelangt eine zunehmende An- 
zahl vorzugsweise der hochfrequenten Schwingungsoszillatoren des Kri- 
stalles in den Grundzustand. Diese Oszillatoren konnen keine Energie 
mehr abgeben und die Linienform wird unsymmetrisch, wenn die Riick- 
stoBenergie nicht groB ist gegen die obere Grenzenergie % co g des Schwin- 
gungsspektrums des Kristalles. Lamb [5] berechnet die Lage und Form 
der Absorptionslinie beim Resonanzeinfang langsamer Neutronen in 
Kristallen und gibt Naherungen fur spezielle Bindungsfalle an. Hier- 
nach besitzt die Absorptionslinie im Fall schwacher Bindung und unter 
der Annahme, daB der Kristall als Debye-Kontinuum aufgefaBt werden 
kann, dieselbe Form wie in einem idealen Gas, wobei jedoch an die Stelle 
der tatsachlichen Temperatur T eine Temperatur T*>T tritt, die 
der mittleren Energie pro Schwingungsfreiheitsgrad des Kristalles ent- 
spricht. Im Falle starkerer Bindung zeigt die Absorptionslinie bei 
tiefen Temperaturen eine komplizierte Struktur und eine Verschiebung 
zu kleineren Energien, verbunden mit dem Auftreten eines Maximums 
an der Stelle der Resonanzenergie. Die Linienform wird um so unsym- 
metrischer, je kleiner das Verhaltnis R/kO ist. 

Die Arbeit von Lamb laBt sich leicht fur die Emissionslinie erweitern 
und kann dann direkt auf die Gammaresonanzprozesse angewandt 
werden. Bei der Resonanzfluoreszenz des 129keV Niveaus in Ir 191 ist 
R = 0,046 eV und £(9 = 0,025 eV. Der Fall schwacher Bindung in der 
Definition nach Lamb [5] ist hier bei Temperaturen T<200°K nicht 
mehr realisiert. 

3. Theorie 

Nach Lamb [5] gilt fur den Wirkungsquerschnitt fur die Resonanz- 
absorption: a[ E ) = [ r^)a W a{ E) (2) 

{T — totale Energiebreite des Resonanzniveaus ; E = Energie der ein- 
fallenden Quanten; a = Wirkungsquerschnitt fur exakte Resonanz). 
W a (E) bestimmt Lage und Form der Absorptionslinie : 

oo 

W a (E) (2/r) Real / dp exp [*> (£-£„+ i 7)2) + g. (fi)] . (3 a) 



Dabei ist 



M = Z 



(p « s ) 2 



X 



(4a) 



2m hco.N 

X [(a s -f 1) ex P (— i ft ft a) s ) + a s exp (ijuH(o s ) — \ — 2aJ 

E ist die Resonanzenergie, oj s die Frequenz der s-ten Normalschwingung 
des Kristalles, m die Kernmasse, p der Impuls des Gammaquants, ~e der 
Polarisationseinheitsvektor, 32V die Zahl der unabhangigen Freiheits- 



105 

128 Rudolf L, Mossbauer: 

grade im Kristall und a 5 die mittlere Besetzungszahl des s-ten Oszillators : 



Wie sich leicht zeigen laBt* gilt fur die Emissionslinie ** : 

oo 

W e (E) = (2/r) Real / dp exp [»> (£ - E + * T/2) + g e (fi)] 
o 
mit 

&M = 2j 2ma J aN x 

1-2S.1. 



(5) 



(3 b) 



(4b) 



X [(oc s + 1) exp (ijuha) s ) + a s exp (— ijuha) s ) 

Die Berechnung der Integrale (3) wird in Debyescher Naherung durch- 
gefuhrt***. Die Debye-Temperaturen der transversalen und der longi- 
tudinalen Schwingungskomponenten werden gleichgesetzt. 

a) Fm in k 6 <Ci folgt 

gaijA ^ — *'/*■? — A* 2 ^ £ > (6a) 

&(/<*) ^ + i/uR—ju 2 Re. (6b) 

ft ist die RiickstoBenergie nach (1) und £ die mittlere Energie pro 
Schwingungsfreiheitsgrad des Kristalles : 



0/ T 



e (D = * r* = 3*r(r/©)»j"( 1 ^- 1 - + |)«<. 

b) Fur,aft6>>1 gilt 



(7) 



«.{<»; fl 






goo(T) 



2£ 



(Pes) 2 l = 



2mha>*N 



a 5 + 



2 J A^O) \ e nmlkT __ x + 2 J 



3N(ha))< 
(kG) 3 



d(hcj) 



e/T 



- (6Rlk&) (Tie)' /(-^y + |)*«. 



(8) 



* Der Beweis sei kurz angedeutet : Die Gl. (8) von Lamb [5] ist fur die Emis- 



sionslinie zu ersetzen durch W({cc s }; {n s }) = 



(«,«>, 



£ + £K)-£(a s ) +tT/2 



wo (a s |H'|w s ) das Matrixelement fiir einen Ubergang n s ->ct s des Gitters bedeutet, 
der von der Emission eines Quants mit dem Impuls p begleitet ist. In Gl. (17) 
von [5] erscheint dann n s an Stelle von a s . Wenn man dann alle folgenden Sum- 
mierungen iiber a s statt iiber n s erstreckt und iiber n s statt iiber a s mittelt, so erhalt 
man unsere Gin. (3 b) und (4 b), wenn man im Endergebnis wieder a s statt n s setzt. 
** Die Funktionen W £ (E) und W a {E) und damit auch die Emissions- und Ab- 
sorptionslinien liegen spiegelsymmetrisch zur Geraden E = E . 

*** Die Summationen in (4) enthalten auch die Summationen iiber die Polari- 
sationsrichtungen. 

•**• Tj nsere Gl. (8) unterscheidet sich um einen Faktor 3 von der bei Lamb [<5] 
angegebenen Gl. (36). 



106 

Kernresonanzfluoreszenz von Gammastrahlung in Ir 191 129 

Der experimentell bestimmbare mittlere Wirkungsquerschnitt o r (T q ; T a ) 
ergibt sich, wenn man a(E) in (2) mit der auf 1 normierten Verteilungs- 
funktion der emittierten (einfallenden) Quanten multipliziert und liber 
alle Energien summiert (T q , T a = Temperatur von Quelle bzw. Ab- 
sorber). Wegen 

oo 

(ri27i) f W e (E) dE = i 



gilt daher 

OO 

o f = fa(E){ri2n)W t {E)dE 

und mit (2) 

d r = (n87i)a JW a (E)W e (E)dE. 





(9) 



Wenn der Ubergang in den Grundzustand auch durch innere Umwand- 
lung (Kon version) erfolgen kann, dann ist zu unterscheiden zwischen 
dem Wirkungsquerschnitt (a ) s fur die Resonanz-Streuung und dem 
Wirkungsquerschnitt (o ) a fur die Resonanz- Absorption. Fur die 
Resonanz-Streuung gilt nach Jackson [6] : 

t n \ ._ (2/q+0 % rr rf _ (2/ fl +1) H „ i , , 

Voh- (27g+ 1} 2n "■ -j* - {2Ig+i) ' 27Z n (1 +a)2 • y™) 

Dabei sind 2l-\-\ die statistischen Gewichte des angeregten (a) und 
des Grundzustandes (g), H die relative Haufigkeit des resonanten 
Isotops, X Q die Resonanzwellenlange, r die totale Energiebreite des 
Resonanzniveaus, T y die partielle Energiebreite fur Strahlungsemission 
und a der Konversionskoeffizient. 

Bei der Resonanzfluoreszenz geht durch Quantenemission nur der 
Bruchteil T y \T aller Zwischenzustande in den Grundzustand iiber, der 
Bruchteil {T — T^\T aber durch Kon version. Bei der Resonanzabsorp- 
tion tragen alle Zwischenzustande zu dem Wirkungsquerschnitt bei und 
es gilt daher 

k). = wr v ) ■ Ms = |^f ^ mm ■ (») 

Im Falle der Resonanzfluoreszenz an Ir 391 (E = 1 29 keV) kann die 
Berechnung der Linienformen bis herab zur Temperatur des fliissigen 
2 naherungsweise in zwei Schritten durchgefuhrt werden: 

1. Im Bereich \E — E \^>T ergeben nur die Werte juk&<\ einen 
wesentlichen Beitrag zu dem Integral (3), und (6) kann noch als brauch- 
bare Naherung beniitzt werden : 



Wj (E) = (2/JT) ftp cos (E -E Q ±R)fi exp (- p T\2 - ju 2 Re) 







(12) 



= (4ir*MS;x) 



wobei 



107 

130 Rudolf L. Mossbauer: 

Dabei gilt fur die Emissions- (e) und Absorptionslinie (a) bzw.* 

Vfe: *«) = (1/2) V»f.exp (-1? 4/4) , (13a) 

Vtf.; *J = (1/2) V^f.exp (-{• 4/4) , (13b) 

* e = (£-£„ 4- .R)/Ar; *.= (£-£,-fi)/|r. (14) 

4=2]/^l2f; J.= 2y«*5 F , (15) 

h = riA e ; £« = riA a . (16) 

Zl ist die Doppler-Breite des Niveaus und T*, T* bezeichnen die Tem- 
peraturen des Absorbers und der Quelle, die nach (7) den mittleren 
Energien pro Schwingungsfreiheitsgrad des Kristalles entsprechen. 

2. Im Bereich um E = E erhalt man eine gute Naherung fiir W(E), 
wenn man den Integrationsbereich in (3) an der Stelle fjikO — \ auf- 
spaltet und in den beiden Bereichen die Naherungen (6) bzw. (8) beniitzt : 

ilk® 
W H (E) = (2/JT) J dju cos [E — E ± R)juexp (— /* JT/2 - p*R e ) + 


00 

+ {21 r) / dp cos (E - E ) 11 exp ( goo (T) - p T/2) . 

1/kO 

Das erste Integral liefert nur einen kleinen Beitrag zu W n (E) und es 
gilt in guter Naherung (rjk&<^l) : 

W Il{ E) = W l{ E) + {E ^f^ jA . (17) 

Im Bereich \E — E \^>T kann der zweite Summand in (17) gegeniiber 
Wj(E) vernachlassigt werden und es folgt durch Zusammenfassen von 
(1 2) und (1 7) fiir die Emissions- bzw. Absorptionslinie : 

W e {E) = (4in f (i e ; x e ) + ^^cSEA— , (18a) 

W a (E) = (4/P)y(f.;«J + {E r^)?™ 2/4 • < 18b ) 

Die durch die Kristallbindung modifizierten Emissions- und Absorp- 
tionslinien des 1 29 keV-LJberganges in Ir 191 besitzen nach (1 8) ** (mit 
Ausnahme in der Umgebung der Resonanzstelle E ) bei der Temperatur T 
nahezu die gleiche Form und Lage wie in einem idealen Gas bei einer 
Temperatur T*, die nach (7) der mittleren Energie pro Schwingungs- 
freiheitsgrad des Kristalles entspricht. An der Resonanzstelle E = E 

*Vgl. Lamb [5], Gl. (32). 
** Die Gin. (18) verlieren ihre Giiltigkeit bei Temperaturen unterhalb der 
Temperatur des fliissigen 2 . 



108 

Kernresonanzfluoreszenz von Gammastrahlung in Ir 191 131 

erhebt sich dariiber bei Temperaturen T<0 eine mit abnehmender 
Temperatur stark ansteigende Linie der Breite r (Fig. 1). Die Resonanz- 
bedingung wird also mit abnehmender Temperatur fur einen immer 
groBer werdenden Bruchteil der Quanten erfullt. 



Emissionsspektrum 
>*750 




E +R Energie 



Fig. 1. Lage und Form der Emissions- und Absorptionslinien des 129 keV-t)berganges in Ir 191 bei T = 88° K, 

fur eine Lebenszeit r = 10~ 10 sec. Nullpunkt der Energieskala unterdruckt; Einheit der Ordinate willkurlich ; 

Hohe der Linien bei E = E (Resonanzenergie) im Vernal tnis 1:150 verkiirzt dargestellt 

Der Wirkungsquerschnitt a r ergibt sich durch Eintragung von (18) 
in (9). Dabei liefern nur zwei Integrale einen wesentlichen Beitrag: 



oo ex. 

f(4iny>(Z e ;x e )(4in W (£ a ;x a )dE und J 



exp[g 00 (r g ) + goo (r a )] 

{{E - £ ) 2 + T2/4] 2 



dE. 



Ausfiihrung der Integrationen ergibt fiir den mittleren Wirkungsquer- 
schnitt fiir die Resonanzabsorption : 



K)a 



]/* 



Pi + A> 



exp 



4R 2 



Al+Al\ 



^ex V [ goo m+ goo (T a )]. (19) 



Die Unscharferelation ergibt fiir die mittlere Lebenszeit r des Resonanz- 
niveaus und fiir die mittlere Lebenszeit r y fiir Zerfall durch Strahlungs- 
emission : 

r = hir= %\T y (1 + a) = r y /(l + a) = 2]/ln 2 (20) 

(a == Konversionskoeffizient, 7] = Halbwertszeit). 

Bei hoheren Temperaturen kann der 2. Term in (19) vernachlassigt 
werden und es folgt nach (11), (19), (20) o ra ~\\x y . Bei tiefen Tempera- 
turen dominiert der von der Lebenszeit unabhangige 2. Term in (19). 



109 

132 Rudolf L. Mossbauer: 

4. Versuchsanordnung 

Fig. 2 zeigt die Versuchsanordnung, Fig. 3 den Aufbau des Absorber- 
Kryostaten. Die Absorber, zwei je etwa 0,4 mm dicke gewalzte Iridium- 
bzw. Platinbleche von 35 mm Durchmesser waren so befestigt, daB bei 
der Abkuhlung eine ungehinderte Kontraktion derselben moglich war. 

Untersucht wurde die Absorption der beim Beta-Zerfall von Os 191 
ausgesandten 129keV Gammastrahlung in Iridium. Fig. 4 zeigt das 
Zerfallschema [7] und das beobachtete Spektrum, das neben der l6d- 
Aktivitat von Os 191 noch Komponenten der 95 d-Aktivitat von Os 185 
enthalt [8]. Die harten, beim iC-Einfang von Os 185 ausgesandten Linien 
von Re 185 bei 640 keV und bei 875 keV durchsetzten die Absorber 

Pb 







70cm 



^-n\$^\\^ 


«t 


m 


^§§^ 


p 


1 


M^§ 




mm 



Fig. 2. Versuchsgeometrie. A Absorber- Kryostat; P Kryostat mit Quelle; D Detektor: NaJ(Tl)-Kristall 

(22 mm hoch, 40 mm Durchmesser) und Photomultiplier; K Kollimator (Bohrung 18 mm); A und P werden 

von den Armen eines schweren Stativs getragen 

nahezu ungeschwacht, wahrend die weiche 129 keV-Strahlung von 
Ir 191 durch den Photoeffekt stark absorbiert wurde. Die harten Kom- 
ponenten von Re 185 uberlagerten daher im Nachweiskristall mit ihrem 
Compton-Kontinuum erheblich die 129 keV-Resonanzlinie. 

Wahrend die Selbstabsorption der Resonanzlinie in der Quelle 
stark mit der Dicke des Praparates zunimmt, erfahren die harten 
Strahlungskomponenten von Re 185 nur eine geringfugige Selbstabsorp- 
tion. Die von der Quelle emittierte Strahlungsintensitat verschiebt sich 
daher mit zunehmender Schichtdicke des Praparates zugunsten der 
harteren Strahlungskomponenten. Die Praparatsubstanz wurde deshalb 
flachenformig angeordnet (Oberflache 80 mm 2 ) und die Menge auf 0,6 g 
beschrankt. Die Substanz — analysenreiner pulverformiger Osmium- 
schwamm — wurde vor der Bestrahlung im Vakuum in eine dunnwan- 
dige Quarzkiivette eingeschmolzen. Die 65 mCurie starke Quelle wurde 
an den Boden eines zur Aufnahme von fliissiger Luft dienenden zylinder- 
formigen Quarzbehalters angeschmolzen, der sich mit dem Praparat 
in einem DewargefaB befand. Die beschriebene Montage der Quelle 
war erforderlich, um Praparatbewegungen infolge einer Kontraktion 
der Aufhangevorrichtung bei der Abkuhlung auf ein Minimum zu be- 
schranken. Derartige Praparatbewegungen konnten zu einer Anderung 
der von der Quelle ,,gesehenen" mittleren Schichtdicke des Absorbers 



110 

Kernresonanzfluoreszenz von Gammastrahlung in Ir 191 133 

fiihren, sofern dieser nicht vollstandig planparallel ist*. Die haupt- 
sachliche experimentelle Schwierigkeit bei der Lebenszeitmessung lag 
in einer sicheren Ausschaltung eines solchen Einflusses von Anderungen 
der Geometrie bei der Abkiihlung auf die Messung. Das Problem wurde 




N 
sec^ 



woo- 



750- 



500- 



250 ~ 



Re w (K a ) 
Ir w (K a ) 



729 keV 
(Ir 191 ) 




— 5,6 sec 

E3 I 0,0¥2 fleV m 
>5-70' 7O sec 



m+E2 



0,729 'MeV 



Jr' 



6V0 keV 




Fig. 3 



675 keV 
(Re m ) 



O 0,7 0,2 0,3 0,¥ 0,5 0,6 0,7 0,6 0,9 1,0 Y\M 
b 

Fig. 4 a u. b 



Fig. 3. Absorber- Kryostat (schematisch) mit einer durch zwei Schnurziige bedienbaren Drehvorrichtung, um 
abwechselnd zwei verschiedene Absorber A 1 und A 2 in den Strahlengang zu bringen. D DewargefaB; S Achse 
des senkrecht zur Zeichenebene verlaufenden Strahlenbiindels ; T Trolitulisolator. Schnurzug und Thermo- 
elemente werden durch die fliissige Luft in Messingrohrchen gefiihrt, die in den Boden des Luftbehalters 

eingelotet sind 

Fig. 4 a u. b. a Zerf allschema von Os 191 . b Spektrum des natiirlichen Osmiumisotopengemisches, gemessen 
hinter 0,4 mm Iridium 4d nach AbschluB der 12d-Neutronenbestrahlung 



befriedigend gelost durch die beschriebene Art der Aufhangung der 
Quelle, durch die Verwendung von nahezu planparallelen Absorbern** 

* Eine Anderung der Schichtdicke des Absorbers von nur 0,1 [i hatte bereits 
zu Absorptionseffekten in der GroGenordnung der gemessenen Effekte gefiihrt. 
** Wir danken der Fa. Heraeus Platinschmelze fur die Herstellung und leih- 
weise Uberlassung der Edelmetallabsorber. 



Ill 



134 



Rudolf L. Mossbauer: 



groBer Oberflache und durch die Wahl eines relativ groBen Abstandes 
der Quelle von den Absorbern (mindestens 50 cm). Die Temperaturen 
der Quelle und der Absorber wurden mit Eisen-Konstantan-Thermo- 
elementen gemessen. Die ganze Anordnung wurde auf optischem Wege 
justiert. 

Fig. 5 zeigt das Blockschema der elektronischen Nachweisapparatur. 
Der Aufbau und die Betriebsweise waren darauf abgestellt, Schwan- 
kungen in der Zahlhaufigkeit auf ein MindestmaB zu beschranken, z.B. 



A 




B 






Ci 




7 




£1 




Fi 






























c 2 




D 2 




5 




Fi 



Fig. 5- Blockschaltbild der elektronischen MeBanordnung. (Die Strom versorgung wurde nicht dargestellt. ) 
A Photomultiplier RCA 6342 Betriebsspannung 1,3 kV; B Vorverstarker [9]; C Hauptverstarker ; D Ein- 
kanaldiskriminator [10]; E Dualuntersetzer Untersetzungsfaktor 2 1 *; F Registrierstufe mit mechanischem 

Zahlwerk 

durch Verwendung von Schaltelementen mit kleinen Temperatur- 
koeffizienten. Die ganze Anlage wurde von einem auf ±0,5% span- 
nungsstabilisierten Netz betrieben. Das Hochspannungsgerat und die 
Heizung der Verstarker wurden zusatzlich durch einen magnetischen 
Spannungsgleichhalter stabilisiert. 



5. Mefiverfahren 
Eine direkte Bestimmung des Wirkungsquerschnittes fur die Kern- 
resonanzabsorption durch eine Messung des totalen Schwachungs- 
koeffizienten ist im allgemeinen nicht moglich, da die Kernresonanz- 
absorption gewohnlich sehr klein ist gegeniiber den Absorptionseffekten 
der Atomhiille. Der Kernresonanzeffekt in Ir 191 wurde daher durch 
eine Differenzmessung bestimmt, wobei die Absorption der Resonanz- 
linie im Resonanzabsorber Iridium bei verschiedenen Temperaturen 
unter Bedingungen gemessen wurde, bei denen die auftretenden Intensi- 
tatsanderungen in direkter Beziehung zu dem Wirkungsquerschnitt 
fur die Resonanzabsorption standen und jegliche Nebeneffekte aus- 
geschaltet wurden. Der Wirkungsquerschnitt a ra ist nach (19) eine 
Funktion der Temperaturen der Quelle und des Absorbers. Aus Grunden 
der Temperaturabhangigkeit der totalen Absorption wurden alle Mes- 
sungen bei konstanter Temperatur des Absorbers durchgefuhrt, d.h. 
es wurde jeweils nur die Temperatur der Quelle variiert. Die Temperatur- 
abhangigkeit der totalen Absorption hat ihren anschaulichen Grund 
darin, daB sich die Zahl der Atome pro cm 2 der Oberflache einer ab- 
sorbierenden Substanz mit der Temperatur andert [1]. Dieser Effekt 
hatte im vorliegenden Fall dem Kernresonanzeffekt entgegengewirkt 



112 



Kernresonanzfluoreszenz von Gammastrahlung in Ir 191 



135 



und ihn dabei vollstandig iiberdeckt, was durch Beschrankung der 
Temperaturvariationen auf die Quelle verhindert wurde. Naturlich 
zeigt auch die Selbstabsorption in der Quelle einen (allerdings kleineren) 
Temperaturgang. Dieser Effekt, der ebenfalls dem Resonanzeffekt 
entgegenwirkt, konnte jedoch experimentell eliminiert werden durch 
abwechselnde Intensitatsmessungen mit dem Resonanzabsorber und 
einem Vergleichsabsorber. Da der Selbstabsorptionseffekt der Quelle 
die Intensitat hinter beiden Absorbern beeinfluBt, wahrend der Kern- 
resonanzeffekt nur beim Resonanzabsorber auftritt, fallt ersterer Effekt 
bei einer Differenzmessung heraus, wenn 
die Absorber so abgestimmt sind, daB 
sie ungefahr gleich stark absorbieren. 

Die Resonanzlinie wurde aus dem 
Zerfallspektrum des Osmium-Isotopen- 
gemisches durch den Einkanaldiskrimi- 
nator D x (Fig. 5) ausgeblendet. Um den 
EinfluB von Schwankungen des Schwel- 
lenwertes des Einkanaldiskriminators auf 
die Messungen zu vermindern, wurde der 
Kanal auf 25 % des ganzen erfaBbaren 
Spektralbereiches geoffnet *. Die aus dem 
Spektrum ausgeblendete Resonanzlinie 
wurde durch entsprechende Wahl der 
Verstarkung der maximalen Kanalbreite 
angepaBt. Die harteren Komponenten des Spektrums fuhrten dabei zu 
einer Obersteuerung des Hauptverstarkers, die jedoch keinen EinfluB auf 
den ausgeblendeten Teil des Spektrums hatte. 

Um die Schwankungen der Verstarkung (insbesondere des Multi- 
pliers) aufzufangen, wurden die Begrenzungen des Kanals so eingestellt, 
daB kleine Schwankungen der Verstarkung in erster Naherung keine 
Anderung der Impulshaufigkeit im Kanal verursachten. Das Ver- 
fahren wird durch Fig. 6 illustriert : Eine Anderung der Verstarkung 
fiihrt zu einer Expansion bzw. Kontraktion des Spektrums langs der 
Abszisse. Wenn der ausblendende Kanal so eingestellt wird, daB 




N(E 2 ) 



— »► /mpu/shdhe. 

Fig. 6. Einstellung der Begrenzungen des 

Kanals des Einkanaldiskriminators zur 

Stabilisierung der Impulshaufigkeit gegen- 

iiber Verstarkungsschwankungen 



N(EJE, 



N{E,)E,._ 



so wird die Impulshaufigkeit, wie sich leicht 



zeigen laBt, in erster Naherung nicht durch kleine Verstarkungsschwan- 
kungen beeinfluBt, weil sich die bei einer Verstarkungsanderung neu 
in den Kanal eintretenden und die austretenden Anteile des Spektrums 
kompensieren. Diese Beziehung ist annahernd erfullt bei der in Fig. 7 
gezeigten bei den Messungen beniitzten Einstellung des Kanals auf die 
Resonanzlinie. Durch das beschriebene MeBverfahren konnten die 



* Hierzu wurde die Gegenkopplung im Fensterverstarker der Schaltung [10] 
erhoht. 



113 



136 



Rudolf L. Mossbauer: 



relativen Schwankungen nicht-statistischer Natur in der Zahlhaufigkeit 
auf rund 0,1 % beschrankt werden. 

Als Resonanzabsorber diente Iridium (Z = 77) , als Vergleichsabsorber 
Platin (Z = 78). Der Intensitatsunterschied der die beiden Absorber 
durchsetzenden Strahlung betrug bei Zimmertemperatur etwa 0,1%. 




— ■- Impulshohe 

Fig. 7. Einstellung des Kanals des Einkanaldiskriminators auf die 129keV-Linie in Ir 191 . Aus 

schnitt aus dem Zerfallspektrum von Os m + Os 191 5d nach der Neutronenbestrahlung. Ausschnitt 

aus dem Zerfallspektrum einer Osmiumquelle, deren Os 181 -Aktivitat bereits abgeklungen ist. Beide Spektren 

sind auf gleiche Intensitat bei hohen Energien normiert. — Das Verhaltnis v der Impulshaufigkeiten der 

129 keV-Linie zu alien im Bereich des Kanals liegenden Linien ist v = 0,757 



Gemessen wurden die totalen Strahlungsintensitaten 7j r und If 1 hinter 
dem Resonanzabsorber (Ir) und dem Vergleichsabsorber (Pt). 

Bei den Messungen zur Bestimmung der Lebenszeit des 1 29 keV- 
Niveaus in Ir 191 befanden sich die Absorber stets auf Zimmertemperatur 
und es wurde nur die Temperatur der Quelle variiert. Bei jeder MeB- 
reihe wurden zunachst wiederholt abwechselnd die Intensitaten I] T (T 2 ) 
und If l (T 2 ) gemessen, wobei sich die Quelle auf der Zimmertemperatur 
T 2 befand. Nach Abkuhlung der Quelle auf die Temperatur Jj des 
flussigen 2 erfolgten analog wiederholte Messungen von I^iTj) und 
JfPi)- J eo ^ e Messung mit dem Resonanzabsorber wurde eingeschlossen 



114 



Kernresonanzfluoreszenz von Gammastrahlung in Ir 191 137 

durch zwei Messungen mit dem Vergleichsabsorber, urn einen linearen 
Gang in der elektronischen Apparatur zu eliminieren. 

Bei den Messungen zur Untersuchung des Einflusses der chemischen 
Bindung auf die Kernresonanzabsorption befanden sich die Absorber 
stets auf der Temperatur des fliissigen 2 . Die Temperatur der Quelle 
wurde variiert zwischen dem Siedepunkt des 2 und dem des Wassers. 

Mit Hilfe des zweiten Zweiges der elektronischen Nachweisapparatur 
(vgl. Fig. 5) wurde parallel zum eigentlichen Experiment taglich das 
gesamte Zerfallspektrum des Osmium-Isotopengemisches aufgenommen. 
Die hierbei gemessenen Spektren dienten zur Ermittlung des (zeit- 
abhangigen) Beitrags des der Resonanzlinie uberlagerten Compton- 
Kontinuums der harten Komponenten von Re 185 zu der Zahlhaufigkeit 
im Kanal des Einkanaldiskriminators. In der Versuchsgeometrie der 
Fig. 2 wurde das Spektrum von Os 185 (95 d) durch Messungen mit einem 
Osmiumpraparat bestimmt, dessen l6d-Os 191 -Aktivitat bereits ab- 
geklungen war. Subtraktion des auf den zeitlichen Aktivitatsabfall 
korrigierten Zerfallspektrums von Os 185 vom Gesamtspektrum Os 185 + 
Os 191 lieferte dann das Zerfallspektrum von Os 191 . Aus diesen Daten 
ergab sich unmittelbar das fur die Auswertung der Messungen benotigte 
zeitabhangige Verhaltnis v der Intensitat der 1 29 keV- Resonanzlinie 
zur gesamten im ausgeblendeten Bereich des Spektrums gemessenen 
Intensitat. Fig. 7 zeigt ein Beispiel. 

6. Auswertung der Messungen 

Im folgenden bezeichnet der Index die 1 29 keV-Resonanzstrahlung, 
der Index i = \,2, ... n die nicht-resonanten Strahlungskomponenten, 
die der Resonanzstrahlung im ausblendenden Kanal tiberlagert sind. 
of' und a] 1 sind die Wirkungsquerschnitte (Einheit cm 2 ) fur die Ab- 
sorption der i-ten Strahlungskomponente in Platin (Pt) bzw. Iridium 
(Ir). n ist die Zahl der Atome pro cm 2 Absorberoberflache und T ist 
die Temperatur der Quelle. Fur die hinter den Absorbern gemessenen 
Strahlungsintensitaten gilt 

I}' (2i) = 2 I]' (23 = 2 I, (rj exp [- n l < ^ (7,)] , (21 a) 



IT ft) = 2 IT ffi) = 2 I.m exp [- » PI af (23] , (21 b) 

t=0 i=0 

I}' (23 = 2 II' (7J = 2 1> (2i) exp [- n" a," (23] , (21 c) 

t=0 i=0 

IT (TJ = 2 IT (23 = 2 1, : (?3 exp [- » pt of (73] . (21 d) 

t=0 i=0 



115 



138 Rudolf L. Mossbauer: 

Da nur die Temperatur der Quelle variiert wird, sind n lT und w pt kon- 
stant und es gelten die Beziehungen 

<yf t (T 2 )=(rt t (T 1 ) = of * * = 0,1,2,...,*, (22a) 

^ r ra = ^ r (r i ) = crJ r * = 1,2,...,*, (22b) 

k; r ra-<rai=i^ra-^rai<iy r . (22c) 

Weiter werden eingefuhrt die relativen Intensitatsanderungen <x t >0, 
die sich beim Temperaturiibergang T 1 -+T 2 (T 2 > 7i) der Quelle infolge 
Anderung der Selbstabsorption in der Quelle ergeben : 

Ii{T 2 ) = (1 + « t ) /.(TJ i = 0, 1, 2, ..., ». (23) 

Die Absorber absorbieren nahezu gleich stark (vgl. Abschnitt 5) und alle 
Temperatureffekte verursachen nur kleine Intensitatsanderungen, d.h. 
es gilt 

|/Pt (r) _ 7 ir (r) | <7 . (r) ,- = 0,1,2,...,*, (24a) 

a^<l »*0, 1,2, ...,*. (24b) 

Fiihrt man als experimentell bestimmbare GroBe die Differenz ein: 
M = lfHT 2 )-I?(T 2 ) _ /Pt^-jjt^ 

/J r (r 2 ) /fcrj 

so erhalt man aus (21) bis (23) 

M ^{2a,/ i (r l ) [exp (- »»o«) - exp(- « Ir aJ r (T 2 ))] - 

- 1,(23 [exp (- n»ol<(T 2 )) - exp (- n^tf))]}//*^) . 

Der erste Term wird nach (24) klein von hoherer Ordnung und es folgt 

if«7,(r I )e«p(-i» fc «f(r 1 ))x l 

xti-ex P (^(Tjr(r 1 )-» Ir of(r 2 ))]/// r (r^. } 

Bezeichnet man mit v den zeitabhangigen Bruchteil, der von der ge- 
samten im Bereich des Kanals des Einkanaldiskriminators gemessenen 
Intensitat auf die Intensitat der Resonanzlinie entfallt (vgl. Abschnitt 5) 

v = /.(T.) exp [- « fc af (rOJ/i^tr.) , (27) 

so folgt aus (26) und (22 c) : 

M^i;n Ir [* fa (r 2 )-tf r J7i)]. (28) 

Bei den Messungen zur Bestimmung der Lebenszeit wurden die Absorber 
nicht gekiihlt. Dann kann der zweite Term in (19) vernachlassigt wer- 
den und der Wirkungsquerschnitt a ra fur die Resonanzabsorption wird 



116 



Kernresonanzfluoreszenz von Gammastrahlung in Ir 191 



139 



umgekehrt proportional zur Lebenszeit r y des Niveaus. Nach (1), (11), 
(15), (19), (20) gilt in diesem Fall: 



2l a + 1 h 2 c* 



(29) 



rj % "]/ m 

~V y \/2kn(T* + T*) X 

X exp [- E 2 j2m c*k(T q * + T*)] . J 

Bei der Auswertung wurden folgende numerische Daten verwendet: 

Spin des Grundzustandes von Ir 191 : I g — f [11] 

Spin des 1 29 keV Niveaus von Ir 191 : I a = § [7] 

Relative Haufigkeit des Isotops Ir 191 : H = }8,5% 

Debye-Temperatur von Iridium: = 285° K [12] 

Fiir die Temperaturen T* und 7^* von Absorber und Quelle folgt aus (7) : 

Zimmertemperatur (3 03° K) : T* = 3 16° K 

Flussige Luft (gemessener Mittelwert 8$° K) : T* = 129° K 



Zahl der Iridium-Atome pro cm 5 



2,78 • 10 21 cm- 2 



7. MeBergebnisse 

Tabelle 1 enthalt die Ergebnisse der Messungen zur Bestimmung 
der Lebenszeit des 1 29 keV-Niveaus in Ir 191 . (Temperaturen der Quelle 



Tabelle 1 



1 


2 


3 


4 


5 


Nr. 


T = T f = 303° K 


T = r, = 88°K 


M-10 3 


{M/v) • 10 3 


1 


1,346±0,31 


1,074 ±0,24 


0,272 ±0,39 


0,341 ±0,49 


2 


4,585±0,19 


4,359 ±0,21 


0,226 ± 0,28 


0,285 ±0,35 


3 


0,580 ±0,25 


0,127 ±0,23 


0,453 ±0,34 


0,574 ±0,43 


4 


0,653 ±0,16 


0,297 ±0,28 


0,356 ±0,33 


0,453 ±0,42 


5 


0,767±0,17 


0,579±0,19 


0,188 ±0,26 


0,240 ±0,33 


6 


0,649 ±0,23 


0,504 ±0,17 


0,145 ±0,29 


0,186 ±0,37 


7 


0,249 ±0,21 


0,166±0,16 


0,083 ± 0,26 


0,108 ±0,34 


8 


0,386 ±0,24 


0,001 ±0,25 


0,385 ±0,35 


0,504 ±0,46 


9 


0,397 ±0,24 


0,038 ±0,20 


0,359±0,31 


0,474 ±0,41 


10 


0,404 ± 0,23 


0,037 ±0,25 


0,367 ±0,34 


0,488 ±0,45 


11 


0,241 ±0,40 


0,052 ±0,26 


0,1 89 ±0,48 


0,254 ±0,64 


12 


0,102 ±0,14 


0,326 ±0,1 7 


— 0,224 ± 0,22 


-0,303 ±0,30 


13 


0,786 ±0,1 5 


0,266 ± 0,26 


0,520 ±0,30 


0,711 ±0,41 


14 


0,960 ±0,29 


0,542 ±0,1 7 


0,418 ±0,34 


0,578 ±0,47 


15 


-1,797 ±0,25 


-1,775±0,20 


— 0,022 ±0,32 


-0,031 ±0,45 


16 


-2,274 ±0,19 


-2,379 ±0,28 


0,105 ±0,34 


0,149 ±0,48 




Ausgeglichen 


er Mittelwert (M/v 


) -10 3 


0,268 ±0,07 



T 1 = 88°K; r 2 = 303°K.) Spalte 1 der Tabelle gibt die Nummer der 
Versuchsserie : Jede Serie best and aus mindestens je 10 Messungen 



117 



140 



Rudolf L. Mossbauer: 



jeder der vier Intensitaten I] x {T^j, If^TJ, Ij T {T 2 ), I^{T 2 ). Bei der 
einzelnen Intensitatsmessung betrug der statistische Fehler 0,04 bis 
0,05 % und die MeBzeit 12 bis 20 min. Taglich wurde eine MeBreihe auf- 
genommen. Um systematische Fehler auszuschlieBen, wurde die Geo- 
metrie bei jeder MeBreihe etwas variiert, durch Drehung der Absorber 



&ra 

[barn] 



I 

%70 



« 



0% 

-1% 

-2% 


jlridium_ jP/atin 

rPlatin 

1 






\ 


1 


\ 


\ 


\ 
















-i"/ 


\ 









60° 160° 2¥0° 320°K 

Tempera fur Tg der Quelle 

a 



80° 



720° 



'■I ?nn° p.w° pa 



_L 



200° 2W° 280° 320° 380°K 
Tempera fur Tg der Quelle 

b 

Fig. 8 a. Relatives Intensitatsverhaltnis (7 Ir — I pt )/J pt der hinter Iridium- bzw. Platinabsorbern gemessenen 
129keV-Gammastrahlung als Funktion der Temperatur der Quelle. Die Temperatur der Absorber betrug 

konstant 88° K 

Fig. 8 b. Der Wirkungsquerschnitt a ra fur die Kernresonanzabsorption in Ir in als Funktion der Temperatur 

der Quelle, fur eine Absorbertemperatur von 88° K. £ MeBpunkte; theoretischer Verlauf fur 

ein quadratisch mit der Frequenz v ansteigendes Schwingungsspektrum (Debye-Spektrum) ; theore- 
tischer Verlauf fur eine mit v 3 proportionale Frequenzverteilung, die bei der gleichen Qrenzfrequenz abge- 
schnitten wurde, d.h. die gleiche Debye-Temperatur besitzt. Die Debye-Temperaturen der transversalen 
und der longitudinal en Komponenten wurden gleichgesetzt 



um ihre Symmetrieachsen, sowie durch kleine Ortsveranderungen 
von Quelle und Absorbern. Der EinfluB dieser Geometrieanderungen 
zeigt sich in den Unterschieden der Intensitatsdifferenzen der Spalten 2 
bzw. 3 der Tabelle 1 . Spalte 4 enthalt die Dif ferenz der MeBwerte der 
Spalten 2 und 3 [vgl. Gl. (25)]. Da die Geometrie bei der einzelnen MeB- 
reihe konstant gehalten wurde, sollten (in guter Naherung) die Werte 
M der Spalte 4 nicht von der Geometrie abhangen und ein Vergleich 
etwa der Spalten 2 und 4 laBt auch keinen EinfluB der Geometrie auf M 



118 



Kernresonanzfluoreszenz von Gammastrahlung in Ir 191 141 

erkennen. Spalte 5 enthalt die auf den Beitrag anderer Linien korrigier- 
ten MeBwerte (M/v) [vgl. Gl. (27)]. Die relativen Schwankungen nicht- 
statistischer Natur der Zahlrate betrugen etwa 0,1% iiber mehrere 
Stunden. Dariiber hinaus zeigte die Zahlrate im Verlauf einiger Tage 
langsame elektronisch bedingte Anderungen von einigen Promille, die 
naturlich keinen Einflufi auf die Messung hatten. Die angegebenen 
mittleren Fehler wurden aus den Schwankungen der Einzelmessungen 
ermittelt. Der Fehler jeder MeBreihe setzt sich zusammen aus den 
Fehlern von vier Intensitatsmessungen [vgl. (25)]. Mit dem Mittelwert 
Wjv der Tabelle 1 folgt nach (28) : 

r y = (3.6± J5) - lO" 10 sec. 

Mit dem Konversionskoeffizienten a = 2,47 [13] ergibt sich die mittlere 
Lebenszeit r des Niveaus zu r = r Y l(i + a) = (1 ,0i J;J) • 10" 10 sec. 

Fig. 8 enthalt die Ergebnisse der Messungen, bei denen die Absorber 
gekiihlt wurden. In Fig. 8a wurden die Differenzen der hinter dem 
Resonanzabsorber und dem Vergleichsabsorber gemessenen Intensi- 
taten I aufgetragen, in Fig. 8 b der daraus mit Hilfe von (28) berechnete 
Wirkungsquerschnitt c ra (T q ) = a ra (T 1 ). (Bezugstemperatur T 2 — 0° C ; 
Lebenszeit r y = 3 ,6 • 1 0~ 10 sec ; Konversionskoef fizient a = 2,47 [13] .) 
Fig. 8 b enthalt neben den MeBpunkten den theoretischen Verlauf des 
Wirkungsquerschnittes nach (19) fur zwei verschiedene Frequenzver- 
teilungen des Schwingungsspektrums von Iridium. 

8. Diskussion 

Die gemessene Lebenszeit ist in Einklang mit der von Sunyar [14] 
angegebenen oberen Schranke von 5 • 1 0~ 10 sec fur die Halbwertzeit des 
Niveaus. Messungen der Konversionskoeffizienten ergaben fur die 
Multipolordnung des 1 29 keV-t)berganges M\-\-E2[15]. Die Werte 
fur das aus den Konversionskoeffizienten der K- und L-Schalen be- 
stimmte Intensitatsverhaltnis MijE2 liegen im Bereich 1,5:1 bis 
5:1 [16], [17], [18]. Das hier gefundene r y ergibt mit diesen Daten f in- 
die t)bergangswahrscheinlichkeit des 1 29 keV-t)berganges einen Ver- 
zogerungsfaktor von mindestens 22 gegeniiber den theoretischen Vor- 
hersagen des Einteilchenmodelles [19] fur einen M l-t)bergang. Der 
£2-t)bergang erfolgt dagegen um mindestens einen Faktor 64 schneller 
als nach dem Einteilchenmodell zu erwarten ware. Der 129keV-Zu- 
stand ist daher als Mehrteilchenzustand aufzufassen. Das Niveau ist 
jedoch vermutlich kein reines Rotationsniveau im Sinne des Kollektiv- 
modelles, denn das Verhaltnis der Energien des 1. und 2. ,,Rotations- 
niveaus" weicht nach [13] um rund 10% von dem durch das Kollektiv- 
modell fur reine Rotationszustande geforderten Wert ab. Tabelle 2 



119 



142 



Rudolf L. Mossbauer: 



enthalt die durch Messungen der Coulomb-Anregung bestimmten re- 
duzierten Ubergangswahrscheinlichkeiten B (E 2) fur die E 2-Komponente 
des 1 29 keV-t)berganges in Ir 191 , sowie die daraus berechneten Lebens- 
zeiten r y (£2)* und Mischungsverhaltnisse M\\ E2. 

Die Mischungsverhaltnisse der Tabelle 2 deuten auf eine starkere 
Bevorzugung der Ml-Komponente hin, als aus den Verhaltnissen der 
Konversionskoeffizienten folgen wiirde [16], [17], [18], doch sind die 
B(£"2)-Werte fur genaue Aussagen noch zu unsicher. 



Tabelle 2 



Literatur 



B(E2)** 
«*cm 4 10+ 60 



Ty{E2) 

(£ = 129keV) 



Mi IE 2 
furT v = 3,6-1O- l0 sec 



US] 

[20] 
[16] 



51 
56 
97 



6,1 • l(T 9 sec 



5,6 
3,2 



10~ 9 sec 
10~ 9 sec 



16:1 

15:1 

8:1 



Die tJbereinstimmung der Messungen an der 129 keV-Gammalinie 
in Ir 191 bei Temperaturen r<200° K mit dem in Debyescher Naherung 
berechneten Verlauf des Absorptionsquerschnittes ist nur qualitativ. 
Die Debyesche Theorie fuhrt in der hier verwendeten Form, bei der die 
Debye-Temperaturen der longitudinalen und der transversalen Kom- 
ponenten gleichgesetzt werden, zu einer Unterdruckung der hoheren 
longitudinalen Komponenten. Eine Berucksichtigung der atomistischen 
Struktur fuhrt jedoch umgekehrt bei Schwingungen mit Wellenlangen 
von der GroBenordnung der Gitterkonstanten wegen der gegentiber dem 
Kontinuum groBeren Tragheit des Gitters zu einer Verschiebung des 
Spektrums zu kleineren Frequenzen. Die Messungen deuten auf eine 
groBere Dichte der Frequenzverteilung des Schwingungsspektrums von 
Iridium bei den hoheren Frequenzen hin, als sie die Debyesche Kon- 
tinuumstheorie liefert. 

Untersuchungen des Verhaltens der spezifischen Warmen ergaben 
bei den meisten Metallen bei tieferen Temperaturen einen Anstieg der 
Debyeschen Temperatur B [21]. Tatsachlich lassen sich die in Fig. 8 
dargestellten Abweichungen der MeBergebnisse von dem nach der 
Debyeschen Theorie zu erwartenden Verlauf des Wirkungsquerschnittes 
ebenfalls durch die Annahme eines Anstiegs von mit abnehmender 
Temperatur erklaren. Zur genaueren Untersuchung sollen die Messungen 
zu tieferen Temperaturen hin ausgedehnt werden. 

* Es gilt die Relation (vgl. [13]) : iJr y (E2) = (^\j-(?±J 2li + 



B{E2), 



21/ +1 

wobei If, 7j = Spin des angeregten und des Grundzustandes, E = Anregungs- 
energie. 

** Die B (£2)-Werte sind nach Angabe der Autoren um einen Faktor 2 unsicher. 



120 

Kernresonanzfluoreszenz von Gammastrahlung in Ir 191 143 

Die Bestimmung der Lebenszeiten von Kernniveaus mittels Ab- 
sorptionsmessungen der hier beschriebenen Art sind infolge der auBer- 
ordentlich kleinen Effekte nur in Ausnahmefallen durchfiihrbar. Da- 
gegen treten bei Untersuchungen der Bindungseigenschaften von Fest- 
korpern bei tiefen Temperaturen urn GroBenordnungen starkere Effekte 
auf. Analoge Bindungseffekte wie bei Ir 191 sind im ganzen Energie- 
bereich weicher Gammastrahlung zu erwarten. Wegen der star ken Ab- 
hangigkeit der Absorptionsquerschnitte von den Schwingungsspektren 
eignet sich die Methode der Resonanzfluoreszenz von Gammastrahlung 
bei tiefen Temperaturen zu Untersuchungen der Frequenzverteilung 
der Schwingungsspektren fester Korper. Die Methode kann daneben 
auch bei der Aufstellung von Zerfallschemas verwendet werden, da die 
starke Resonanzabsorption bei tiefen Temperaturen nur bei solchen 
Linien auftritt, die einem Ubergang in den Grundzustand entsprechen. 
Die Untersuchungen werden fortgesetzt. 

Es ist mir ein Anliegen, Herrn Professor H. Maier-Leibnitz fur die Anregung 
zu dieser Arbeit, fur interessante Diskussionen und freundliche Forderung herzlich 
zu danken. Herrn Professor J. H. D. Jensen danke ich fur aufschluBreiche Dis- 
kussionen. Herrn Professor W. BoTHE(f) und Herrn Professor K. H. Lauter- 
jung danke ich dafiir, daB sie die Durchfuhrung dieser Arbeit am Max-Planck- 
Institut fur medizinische Forschung in Heidelberg ermoglicht haben. 

Literatur 

[1] Malmfors, K. G.: Ark. Fysik 6, 49 (1953)- — [2] Malmfors, K. G. in: 
K. Siegbahn, Beta- and Gammaray Spectroscopy. Amsterdam 1955- — [3] Metz- 
ger, F. R. : Report at the Glasgow Conference 1954, S. 201. — J. Franklin Inst. 
261, 219 (1956). - Phys. Rev. 101, 286 (1956); 103, 983 (1956). - Schopper, H.: 
Z. Physik 144, 476 (1956). - Swann, C. P., and F. R. Metzger: J. Franklin 
Inst. 261, 667 (1956). - [4] Franz, W.: Z. Physik 98, 314 (1936). - [5] Lamb jr., 
W. E.: Phys. Rev. 55, 190 (1939). Siehe auch Steinwedel, H., u. J. H. Jensen: 
Z. Naturforsch. 2a, 125 (1947). — [6] Jackson, J.D.: Canad. J. Phys. 33, 575 
(1955)- — [7] Mihelich, I. W., M. McKeown and M. Goldhaber: Phys. Rev. 
96, 1450 (1954). — [8] Marty, N., et M. Vergnes: J. Phys. Radium 18, 233 
(1957)- - [9] Foote, R. S., and H. W. Koch: Rev. Sci. Instrum. 25, 750 (1954). - 
[10] Johnstone, C. W. : Nucleonics 11, No, 1, 36 (Jan. 1953). — [11] Murakawa, 
K., and S. Suwa: Phys. Rev. 87, 1048 (1952). — [12] Seitz, F.: The Modern 
Theory of Solids, S. 110. 1940. — [13] Davis, R. H., A. S. Divatia, D. A. Lind 
and R.D.Moffat: Phys. Rev. 103, 1801 (1956). — [14] Sunyar, A. W.: Phys. 
Rev. 98, 653 (1955). — [15] Swan, I. B., and R.D.Hill: Phys. Rev. 88, 831 
(1952). - [16] Bernstein, E. M., and H. W. Lewis: Phys. Rev. 105, 1524 (1957)- 
[17] Mihelich, I. W., and A. de-Shalit: Phys. Rev. 93, 135 (1954). - [18] McGo- 
wan, F. K.: Phys. Rev. 93, 163 (1954). — [19] Blatt, I. M., and V. F. Weiss- 
kopf: Theoretical Nuclear Physics. New York 1952. — [20] Huus, T., I. H. 
Bjerregaard and B. Elbeck: Dan. Mat. Fys. Medd. 30, Nr. 17 (1956). — 
[21] Leibfried, G., u. W. Brenig: Fortschr. Phys. 1, 187 (1953). 



KERNRESONANZFLUORESZENZ 

VON GAMMASTRAHLUNG IN Ir 191 1 

(Nuclear Resonance Fluorescence 

of Gamma Radiation in Ir 191 ) 



3. THEORY 

According to Lamb 5 $ the cross section for resonance absorption 
is 

a(E)=(r 2 /4)a W a (E) (2) 

where r= total width of the resonance level, E = energy of the inci- 
dent photons, and a = cross section at resonance. 

The position and shape of the absorption line is given by W a (E): 



a 
where 



W a (E) = (2/D Real / dpi exp [i/x(E - E + ir/2) + g a (/i)] (3a) 



, y (p-e s ) 2 



x [(a s + l) exp(-ip.Hw s ) + Q? s exp (ip.tiw s ) - 1 - 2a s ] (4a) 

In these equations E is the resonance energy, co s the frequency of 
the s-th normal mode of the crystal, m the nuclear mass, p the mo- 
mentum of the photon, e the unit polarization vector, 3N the number 
of degrees of freedom of the crystal, and ~a s the average occupation 
number of the s-th oscillator, 

a s = l/[exp (nu> s /kT) - 1] (5) 

As can easily be shown, § 



t Translation of Sec. 3 of the article from Z. Physik, 151, 124 
(1958) which is reproduced in its entirety preceding this translation. 

|For references see the German original. 

§The proof may be briefly indicated: For the emission line, Eq. (8) 
of Lamb 5 must be replaced by 

(a s |H'|n s ) 



W({a s };{n s })- 



E -E + E(n s ) - E(a s ) + ir/2 
121 



122 THEORY 

one obtains for the emission linet 

oo 

W e (E) = (2/D Real / d M exp [i/Lt(E - E + ir/2) + g e (/i)] (3b) 

o 

where 

V 1 (P«e s ) 2 



g e (M) f 



*■ — ■ 2mfia> s N 
s 



x [(a s + 1) exp (i/iliaj s ) + a s exp (-i/i"hw s ) - 1 - 2a s ] 

(4b) 

The integral in (3) will be evaluated in the Debye approximation 
with the Debye temperatures of the transverse and longitudinal modes 
set equal to each other. I 

(a) For /ik0<^l one obtains 

g a (/x)» — i/xR - \i 2 Rz 

g e (M)«+i/uR - jLt 2 Rc (6a) 

Here R is the recoil energy as given by (1) (see the German original, 
p. 126), and e is the average energy per vibrational degree of free- 
dom of the crystal: 

e/ r T f 1 l\ 

e(T) = kT* = 3kT(T/0) 3 J - T ^— + ± t 3 dt (7) 

o \e l - 1 ^/ 

(b) For /xk0^> 1 one has 



where (a s |H'|n s ) is the matrix element for a lattice transition n s 
— - a s accompanied by the emission of a photon of momentum p. In 
Eq. (17) of Ref . 5, a s must be replaced by n s . Extending all the fol- 
lowing sums over a s instead of n s and averaging over n s instead of 
a s , one obtains (3b) and (4b), provided a s is substituted for n s in the 
final relation. 

fThe emission and absorption lines W e (E) and W a (E) are mirror 
images of each other with respect to E = E . 

JThe summations in (4) also extend over the polarization direc- 
tions. 



THEORY 123 



(P*e s ) 



ge<-; T) = g a (-; T) = g„(T) = -2 £ ^£n &* + (1 / 2 » 

s s 

-- 2R / nss^at./sJ a^r- d(t!a;) 

0/T / 1 1 \ 

= -(6R/k0)(T/0) 2 / f— pi — + ± t dt (8)t 

To get the experimentally determined average cross section 
<J r (T q ;T a ), where T q and T a are the source and absorber tempera- 
tures, respectively, ct(E) in (2) is multiplied by the normalized dis- 
tribution function W e (E) of the emitted (incident) photons, and the 
product integrated over energy. That is, 

oo 

a r = f o (E) (r/27r)W e (E) dE 
6 

and, using (2), 

oo 

<f r = (r 3 /87r)a / W a (E)W e (E) dE (9) 

where 



(1/2*)/ W e (E)dE = 1 
o 

If the transition to the ground state can also take place by internal 
conversion, one must differentiate between the cross section (a ) s 
for resonance scattering and the cross section (a ) a for resonance 
absorption. For resonance scattering, according to Jackson, 6 

2I a + 1 A 2 it _ 2I a + 1 A 2 1 

(cr °>s " 2i^TT ^T H r* " SgTl 2i H (T+~5p (10) 

Here 21+1 are the statistical weights of the excited state a and the 
ground state g, H the relative abundance of the resonantly absorbing 
isotope, A the resonance wavelength, r the total energy width of the 
resonance level, T y the partial energy width for radiative transition, 
and a the conversion coefficient. 

In the case of resonance fluorescence, only the fraction T y /T of 



a VJ-./vv e i 

o 



tOur Eq. (8) differs from Eq. (36) of Lamb 5 by a factor of 3. 



124 THEORY 



all intermediate states proceeds to the ground state by photon emis- 
sion, the fraction (r - T )/T proceeding through conversion. In the 
case of resonance absorption, all intermediate states contribute to 
the cross section, and we have 

r 2i a + l a 2 r v 

K)a=f r K)s=^TT i Hf (11) 

For resonance fluorescence in Ir 191 (E = 129 kev), the line shape 
down to liquid-oxygen temperature can be calculated using two dif- 
ferent approximations. 

1. In the region |E - E |^> T, only values of /xk0<l yield a sub- 
stantial contribution to the integral in (3), and (6) is still a useful ap- 
proximation: 



Wj(E) = (2/D / d/i cos(E-E ± R) ju exp(-jur/2 - ^i 2 Rz) 

o 

= (4/r 2 )^;x) (12) 

Here, for the emission line 

0(«e;xe) = ti/2)Yv ( e exp(-||x 2 e /4) (13a) 

while for the absorption line 

^a;* a ) = (l/2)l/5T la exp(-| 2 a x 2 a /4)t (13b) 

In the above equations 

x e = (E - E + R)/(l/2)r x a = (E - E - R)/(l/2)r (14) 



A e = 2VRkTq A a = 2KRkT*/ (15) 

£e = r/A e | a = T/A a (16) 

The quantity A is the Doppler width of the level; T£ and Tg denote, 
respectively, the effective absorber and source temperatures, which, 
according to (7), correspond to the average energy per vibrational 
degree of freedom. 

2. In the region near E = E , a good approximation for W(E) is 
obtained by splitting the integral in (3) at p.k0 = 1 and using approxi- 
mations (6) and (8) in the regions of their validity. Thus 



tSee Lamb, 5 Eq. (32). 



THEORY 125 

l/k0 

W n (E) = (2/r) J d/i cos (E - E ±R) /Li, exp (-/jT/2- p. 2 R C ) 



+ (2/D f djLt cos (E - E ) /x exp [ goo (T) - /xr/2] 
1/ke 

The first integral makes only a small contribution to Wji(E), so that 
for (I/kO) <$C 1 one obtains as a good approximation, 

exp[g w (T)] 
W n (E) = W X (E) + (E I Eq)2 + r2/4 (17) 

In the region |E - E |^> T the second term in (17) is negligible com- 
pared to Wj(E). Combining (12) and (17), the energy distributions of 
the emission and absorption lines become 

w e (E) = (Vil * (S e ;x e ) + (E e ? E [ o g ; ( + T rV4 < 18a > 

and 

exp[g 00 (T a )] 

w a (E) = (4/r 2 ) ^(| a ;x a ) + (E _ Eo)2 + r y4 (18b) 

According to (18), f the crystal binding modifies the emission and 
absorption lines of the 129-kev transition in Ir 191 in the following 
way: Except for the region near resonance, at a temperature T, the 
lines have almost the same shape and positions as they would have in 
an ideal gas at a temperature T*. As shown in (7), T* corresponds 
to the average energy per vibrational degree of freedom of the crys- 
tal. For temperatures T<0 an additional line of width r, whose in- 
tensity increases greatly with decreasing temperature, appears at the 
resonance energy E = E (Fig. 1; see the German original). Thus, 
the fraction of photons satisfying the resonance condition increases 
with decreasing temperature. 

The cross section a r is obtained by substituting (18) into (9). Only 
two of the integrals yield an appreciable contribution: 

oo 

/(4/I*)*(«e;*e)(Vl*)*(* a ; x a) dE and 



7 expfeooCTql + gooCTa)] 
J [(E-E ) 2 + r 2 /4] 2 dE 

t Equations (18) lose their validity at temperatures below liquid 
oxygen temperature. 



126 



THEORY 



The average cross section for resonance absorption obtained by inte- 
gration of these expressions is 



K)a 



'ra 



2 |/A 2 a +A| 



exp 



4R 2 



A 2 a + A|_ 



K>; 



x exp[g 00 (T q ) + g c0 (T a )] 
According to the uncertainty relation, 

t =Vr = Vr y (l + a) = T y /(1 +a) = T 1/2 /ln 2 



(19) 



(20) 



where r is the mean life and T x / 2 is the half- life of the resonance 
level, T y is the partial life for y decay, and a is the conversion co- 
efficient. 

At higher temperatures one can neglect the second term in (19), 
and from (11), (19), and (20) it follows that a ra ~ 1/t^. At low tem- 
peratures the second term in (19), which is independent of lifetime, 
dominates. 



NUCLEAR RESONANCE ABSORPTION 
OF GAMMA RAYS IN Ir 19l t 



Under normal conditions, it is difficult to observe the resonance 
fluorescence of nuclear gamma rays because of the recoil energy lost 
by the photons during both emission and absorption. In general, these 
losses are sufficient to appreciably displace the emission and absorp- 
tion lines, thereby destroying the resonance condition. It is possible 
to compensate for the recoil energy losses by, for example, moving 
the source with respect to the absorber with the aid of an ultracentri- 
fuge, 1 or by increasing the thermal motion of the emitting and absorb- 
ing atoms. 2 In performing experiments of the latter type, we found 
that at low temperatures, contrary to what was expected, a pronounced 
increase in resonance absorption occurred. 3 With the help of a theory 
developed by Lamb, 4 this effect was attributed to the fact that in solids 
the recoil momentum does not always produce a change in the vibra- 
tional state of the crystal lattice. Instead, for a fraction of the gamma 
transitions, the solid as a whole can take up the recoil momentum. 
Thus, according to this theory, the emission and absorption spectra 




10 cm 



Fig. 1. Experimental arrangement: The detector D ac- 
cepts only photons emitted by the source while 
moving on the solid portion of the path shown. 



t Translation of article in Naturwissenschaften, 45, 538 (1958). 

127 



128 



NUCLEAR RESONANCE ABSORPTION 



contain very strong lines of natural width superimposed upon a broad 
distribution resulting from the thermal motion of the atoms bound in 
the crystal lattice. Because of the vanishingly small recoil energy 
losses, these lines appear undisplaced at the resonance energy posi- 
tion (i.e., at the excitation energy of the nuclear level under investi- 
gation). 

We have now demonstrated the existence of these unshifted reso- 
nance lines by means of a "centrifuge" method, employing velocities 
of only a few centimeters per second. The experimental arrangement 
is shown in Fig. 1, while Fig. 2 shows the results obtained for the 





-0.2 

-0.4 

-0.6 

-0.8 

-1.0 

-1.2 
Fig. 2, 



jJr.jPt 
jPt 

-2xl0 _s -lxl0" s 1x10" 


AE *• 

2xl0~ 5 3xl0" 5 4xl0 -5 5xl0~ 5 ev 


1 1 | 1 

-4-2 2 


i i i i ■ 
4 6 8 10 T 12 cm/sec 


1 1 | 1 

V 


i . . . j. 



Difference in intensity of the 129-kev gamma 
transition in Ir 191 measured behind a resonance 
absorber (iridium) and a comparison absorber 
(platinum). This intensity difference is plotted as 
a function of v, the relative velocity of the source 
with respect to the absorber. Both source and ab- 
sorber are at a temperature of 88 °K. aE = (v/c)E 
denotes the energy shift of the 129-kev gamma 
rays. 



129-kev transition in Ir 191 . Thus, a new method for the direct deter- 
mination of the level widths of low- lying excited nuclear states has 
been found. In our case, the line width of the 129-kev level in Ir 191 
agrees, within the limits of error, with the value 6.5 x 10" 6 ev deter- 



. 



NUCLEAR RESONANCE ABSORPTION 129 

mined previously, using a less direct approach. 3 This width corre- 
sponds to a lifetime of 1.0 x 10" 10 sec. 

Laboratorium fiir Technische Physik der Technischen Hochschule, 
Miinchen, und Max- Planck- Institut fiir Medizinische Forschung, 
Heidelberg. 

RUDOLF L. MOSSBAUER 

Received August 13, 1958 

REFERENCES 

1. a P. B. Moon, Proc. Phys. Soc. (London), A64, 76 (1951). b W. G. 
Davey and P. B. Moon, Proc. Phys. Soc. (London), A66, 956 (1953). 

2. a K. G. Malmfors, Ark. Fysik, 6, 49 (1952). b F. R. Metzger and 
W. B. Todd, Phys. Rev., 95, 853 (1954). C F. R. Metzger, Phys. 
Rev., 97, 1258 (1955); J. Franklin Inst., 261, 219 (1956). d H. Schop- 
per, Z. Physik, 144, 476 (1956). 

3. R. L. Mossbauer, Z. Physik, 151, 124 (1958). 

4. W. E. Lamb, Jr., Phys. Rev., 55, 190 (1939). 



130 



KERNRESONANZABSORPTION VON y-STRAHLUNG IN Ir 1 " 



211 









Kernresonanzabsorption von y-Strahlung in Ir 191 

Von Rudolf L. Mossbauer 

Aus dem Laboratorium fiir technische Physik der Technischen Hochschule in Miinchen 
und dem Institut fiir Physik im Max-Planck-Institut fiir medizinische Forschung in Heidelberg 

(Z. Naturforschg. 14 a, 211—216 [1959] ; eingegangen am 5. November 1958) 

Bei der Emission und Selbstabsorption von weicher y-Strahlung in Kernen treten bei tiefen Tem- 
peraturen in Festkbrpern sehr starke Linien mit der naturlichen Linienbreite auf. Diese Linien er- 
scheinen als Folge davon, daB bei tiefen Temperaturen bei einem Teil der Quanteniibergange der 
;'-RiickstoBimpuls nicht mehr vom einzelnen Kern aufgenommen wird, sondern von dem Kristall als 
Ganzes. Da die scharfen Emissions- und Absorptionslinien energetisch an der gleichen Stelle liegen, 
tritt ein sehr starker Resonanzfluoreszenzeffekt auf. Durch eine „Zentrifugen"-Methode, bei der die 
Emissions- und Absorptionslinien gegeneinander verschoben werden, lafit sich der Fluoreszenzeffekt 
unterdriicken und so eine unmittelbare Bestimmung der naturlichen Linienbreite von Resonanzlinien 
vornehmen. Erste Messungen nach dieser Methode ergeben fiir die Lebenszeit x des 129 keV-Niveaus 
inlr»": T=(l,4 lj;f) -10"" sec. 



Die Methode, Kernniveaus durch Einstrahlung 
der eigenen y-Linie 1 zur Fluoreszenz anzuregen, 

1 In besonderen Fallen ist eine Fluoreszenzanregung durch 
Einstrahlung eines Quantenkontinuums moglich : J. E. Dra- 
per u. R.L.Hickok, Phys. Rev. 108, 1280 [1957]. - E. 
Haywakd u. E.G. Foli.br, Phys. Rev. 106, 991 [1957]. 



wird in zunehmender Weise zur Bestimmung der 
Lebenszeiten x kurzlebiger Kernzustande (t<10 -10 
sec) verwendet. 

Die Kernresonanzfluoreszenz von y-Strahlung ist 
unter normalen Bedingungen nur schwer zu beob- 
achten. weil die y-Quanten bei ihrer Emission und 



131 



212 



R. L. MOSSBAUER 



Absorption infolge Abgabe von RiickstoBimpuls an 
die emittierenden und absorbierenden Kerne so 
hohe RiickstoBenergieverluste erleiden, daB die 
Emissions- und Absorptionslinien erheblich gegen- 
einander verschoben werden und daher die Reso- 
nanzbedingung verletzt wird. Es gibt im wesent- 
lichen drei Methoden, durch Kompensation der 
RiickstoBenergieverluste meBbare Werte fiir den 
Wirkungsquerschnitt fiir die Kernresonanzfluores- 
zenz zu erzielen: 

1. DoppLER-Verschiebung der Quantenenergien 
durch mechanische Bewegung der Kerne mit Hilfe 
von Ultrazentrif ugen 2 . 

2. DoppLER-Verbreiterung der Emissions- und 
Absorptionslinien durch Temperaturerhohung, um 
die Uberdeckung der beiden Linien zu verbessern 3 . 

3. DoppLER-Verbreiterung oder DoppLER-Verschie- 
bung der Quantenenergien durch einen fruheren 
Emissions- oder AbsorptionsprozeB, z. B. einen dem 
y-Ubergang vorangehenden /?-t)bergang oder einen 
Teilcheneinfang 4 . 

In der vorliegenden Arbeit wird iiber eine Me- 
thode berichtet, bei der das Auftreten der RiickstoB- 
energieverluste verhindert wird und die Resonanz- 
bedingung daher nicht verletzt wird. Das beschrie- 
bene Verfahren dient zur unmittelbaren Messung 
der Lebenszeiten niedriger, angeregter Kernzustande. 
Erste Messungen an dem 129 keV-Ubergang in Ir 191 
werden mitgeteilt. 

1. Grundlagen der Mefimethode 

In einem fruheren Experiment 5 wurde bei tiefen 
Temperaturen im Gegensatz zur klassischen Erwar- 
tung ein starker Anstieg der Kernresonanzabsorp- 



tion bei dem 129 keV-Niveau in Ir 191 beobachtet. 
Dieser Effekt wurde mit Hilfe einer Theorie von 
Lamb 6 als Folge der Kristallbindung der Absorber- 
und Praparatsubstanzen gedeutet und ist in Fest- 
korpern allgemein bei tiefen Temperaturen und 
weicher y-Strahlung zu erwarten. 

Die Emission oder Absorption eines Quants durch 
einen in einem Kristall gebundenen Kern fiihrt im 
allgemeinen zu einer Anderung des Schwingungs- 
zustandes des Kristallgitters, das den RiickstoB- 
impuls aufnimmt. Wegen der Quantelung der inne- 
ren Energie kann der Kristall die RiickstoBenergie 
nur in diskreten Betragen aufnehmen. Mit abneh- 
mender Temperatur nimmt die Wahrscheinlichkeit 
fiir die Anregung der inneren Niveaus immer mehr 
ab, weshalb bei weicher y-Strahlung 7 bei einem 
Teil der Quantenubergange der Kristall als Ganzes 
den RiickstoBimpuls aufnimmt. Die hierbei emittier- 
ten bzw. absorbierten Quanten erleiden wegen der 
groBen Masse des Kristalles praktisch keine Energie- 
verluste und erfiillen ideal die Resonanzbedingung. 

Abb. 1 zeigt die theoretischen Emissions- und Ab- 
sorptionsspektren des 129 keV-Uberganges in Ir 191 
bei einer Temperatur von 88 °K. 

Die Spektren enthalten je zwei Anteile: 

1. Eine breite, die thermische Bewegung der im 
Kristallgitter gebundenen Atome widerspiegelnde 
Verteilung. Die in den Bereich dieser „thermischen 
Linie" fallenden Quantenubergange sind mit einer 
Anderung des Schwingungszustandes des Kristall- 
gitters gekoppelt. 

2. Eine auBerordentlich starke Linie mit der na- 
turlichen Linienbreite, die die Quantenubergange 
enthalt, bei denen kein RiickstoBenergieverlust auf- 
tritt, weil der Kristall als Ganzes den RuckstoB- 
impuls aufnimmt. Diese „riickstoBfreie Linie" er- 



2 P. B. Moon, Proc. Phys. Soc, Lond. A 64, 76 [1951]. - 
P. B. Moon u. A. Storruste, Proc. Phys. Soc, Lond. A 66, 
585 [1953]. - W. G. Davey u. P. B. Moon, Proc. Phys. 
Soc, Lond. A 66, 956 [1953]. - F. R. Metzger, J. Frank- 
lin Inst. 261, 219 [1956]. - V. Knapp, Proc. Phys. Soc, 
Lond. A 70, 142 [1957]. 

s K. G. Malmfors, Ark. Fysik 6, 49 [1953]. - F. R. Metzger 
u. W.B.Todd, Phys. Rev. 95, 853 [1954]. - F. R. Metz- 
ger, Phys. Rev. 97, 1258 [1955] ; 98, 200 [1955]. - F. R. 
Metzger, J. Franklin Inst. 261, 219 [1956]. - H. Schopper, 
Z. Phys. 144, 476 [1956]. 

4 K.Ilakovac, Proc. Phys. Soc, Lond. A 67, 601 [1954]. - 
F. R. Metzger, Report at the Glasgow Conference 1954, 
S.201; Phys. Rev. 101, 286 [1956]; 103, 983 [1956]; 
110, 123 [1958]. - H. Schopper, Z. Phys. 144, 476 [1956]. 
— C. P. Swann u. F. R. Metzger, Phys. Rev. 108, 982 
[1957]. — S. S. Hanna u. L. Meyer-Schutzmeister, Phys. 



Rev. 108, 1644 [1957]. - L. Grodzins, Phys. Rev. 109, 
1014 [1958]. - V. Knapp, Proc. Phys. Soc, Lond. 71, 
194 [1958]. - P. B. Smith u. P. M. Endt, Phys. Rev. 110, 
397, 1442 [1958]. - F. R. Metzger, C. P. Swann u. V. K. 
Rasmussen, Phys. Rev. 110, 906 [1958]. - V. K. Rasmussen, 
F. R. Metzger u. C. P. Swann, Phys. Rev. 110, 154 [1958]. 
— B. Duelli u. L. Hoffmann, Z. Naturforschg. 13 a, 204 
[1958]. - G. M. Griffiths, Proc. Phys. Soc, Lond. 72, 
337 [1958]. 
s R. L. Mossbauer, Z. Phys. 151, 124 [1958]. 

6 W. E. Lamb jr., Phys. Rev. 55, 190 [1939]. 

7 Bei harter y-Strahlung ist die RiickstoBenergie groB gegen 
die obere Grenzenergie des Schwingungsspektrums des 
Kristalles und es ist eine ungehinderte Aufnahme der Riick- 
stoBenergie in Form von innerer Energie durch den Kristall 
moglich. 



132 



KERNRESONANZABSORPTION VON y-STRAHLUNG IN Ir 191 



213 



Emissionsspektrum 
x2O0 




I(v) 



Energie 



Abb. 1. Lage und Form der Emissions- und Absorptionsspek- 
tren des 129 keV-t)berganges in Ir 191 bei T = 88 °K fiir eine 
Lebenszeit t = 1,4-10 -10 sec. Nullpunkt der Energieskala 
unterdriickt; Einheit der Ordinate willkiirlich; Hone der Li- 
nien bei E=E (Resonanzenergie) im Verhaltnis 1 : 200 ver- 
kiirzt dargestellt. 

scheint daher in Emission und Absorption unver- 
schoben an der Stelle der Resonanzenergie E . 

In der vorliegenden Arbeit wurden am Beispiel 
von Ir 191 die „riickstoBireien" scharfen Emissions- 
und Absorptionslinien mit Hilfe einer „Zentrifu- 
gen"-Methode nachgewiesen. Dabei wurde die Quelle 
gegen den Absorber bewegt, wobei durch den Dopp- 
LER-Effekt die Emissionslinie nach groBeren oder 
kleineren Energien verschoben wurde. Durch diese 
Verschiebung der Emissionslinie wurde die bei ru- 
hender Quelle vorhandene vollstandige Uberdeckung 
der „riickstoBfreien" Emissions- und Absorptions- 
linien aufgehoben. Damit wurde die Resonanzbedin- 
gung verletzt und der starke Resonanzabsorptions- 
effekt der „riickstoBireien u Linien zum Verschwin- 
den gebracht. Eine Analyse der den Resonanzabsor- 
ber (Iridium) durchsetzenden 129 keV-y-Strahlung 
von Iridium als Funktion der Relativgeschwindigkeit 
von Quelle und Absorber lieferte dann unmittelbar 
die Breite der „ruckstoBfreien" Linien, d. h. die 
natiirliche Linienbreite und damit auch die Lebens- 
zeit des 129 keV-Niveaus von Ir 191 . 

2. Theorie 

Wenn die Quelle mit der Geschwindigkeit v in 
Richtung auf den Resonanzabsorber bewegt wird, 
dann gilt 8 fiir die Intensitat / der Resonanzstrahlung 
hinter dem Resonanzabsorber 9 : 



■ const /> e (E + 1. E \ e-°( £ )-" dE . 



(1) 



mit 



n ist die Zahl der Atome des resonanten Isotops pro 
cm 2 Absorberflache und E ist die Resonanzenergie. 
W e (E) ist die Energieverteilung der Quanten einer 
ruhenden Quelle und a(E) ist der Wirkungsquer- 
schnitt fiir die Resonanzabsorption: 

0(£).-(r»/4)o,r.(£) (2) 

Dabei sind / a und / g der Spin des angeregten und 
des Grundzustandes, A die Resonanzwellenlange, 
r die totale Energiebreite des Resonanzniveaus und 
r v die partielle Energiebreite des Resonanzniveaus 
fiir Strahlungsemission. 

Die Absorptionslinie ist nach Lage und Form ge- 
geben durch 

CO 

W & (E) = (2/r) Real/d/* exp [i iu(E-E + i JT/2) 

+ ga(/0]- (4a) 
Dabei ist 

*W"'?i^3rIft +1 » (4b) 

• exp ( — ijuh <o s ) +a, exp (i ju h co s ) — 1 - 2 a s ] . 

E ist die Resonanzenergie, a> s die Frequenz der 
5-ten Normalschwingung des Kristalles, m die Kern- 
masse, p der Impuls des y-Quants, ^T der Polarisa- 
tionseinheitsvektor, 3 A^ die Zahl der unabhangigen 
Freiheitsgrade im Kristall und a s die mittlere Be- 
setzungszahl des 5-ten Oszillators. 

Im Fall der Emissionslinie [W e (E)] ist g a (yu) in 
(4 a) zu ersetzen durch 



ge(/") - 2 



(pes)' 



2 m h a>s N 
+ a. s exp(-ijuho) s ) -1 



[(a s + l) exp(i>feco s ) 

2 a,]. (5) 



Ferner ist 



fw & (E)dE = fW e (E)dE = 2n/r. (6) 
o o 

Fiir ju h a)g > 1 gilt (h a> g = obere Grenzenergie des 
Schwingungsspektrums) : 



8 Fiir Einzelheiten der Ableitung wird auf Anm. 5 i • verwie- 
sen. 

9 Die Absorption der Hiillenelektronen kann im Bereich der 
Resonanzlinie als unabhangig von der Energie angenom- 
men werden. 



133 



214 



R. L. MOSSBAUER 



ge(°o;D=g a (oo;r) (7) 

Wenn der Wirkungsquerschnitt fiir die Resonanz- 
absorption vorzugsweise durch die „rucksto8freien" 
Linien bestimmt wird und wenn zwischen der Niveau- 
breite J" und der oberen Grenzenergie hw g des 
Schwingungsspektrums die Ungleichung T <^ k <o g 
besteht, dann gilt in der Umgebung der Resonanz- 
stelle E in guter Naherung (T q , T a = Temperatur 
von Quelle bzw. Absorber) : 



W e (E) = (2/r) fdju cos (£-£„) fx 
6 

•exp[ goo (T)-f*r/2] 

und o(E) = (r 2 /4) o expg °° (r ^_ 

Damit folgt fiir die Strahlungsintensitiit hinter dem 
Resonanzabsorber bei einer Verschiebung der Emis- 
sionslinie von der GroBenordnung der natiirlichen 
Linienbreite aus (1), (8) und (9): 



exp goo (rq ) 
(E-E )* + r*/4 

(9) 



I(v) — const / 



exp goo (r q ) 



[£+(t;/c)£ -£: ]*+rV4 



(10) 



exp 



d£. 



1+[(2ID(E-E )]* 
I(v) ist eine symmetrische Funktion. 

Im Fall „schwacher Absorption" 10 , d. h. fiir 
na expg oo (T !l )<l folgt aus (10) [mit (E - E ) / 
(r/2)=xund(v/c)Ej(r/2)=y]: 

/H = ^ L expg 00 (7' q ) (11) 



r/2 



dx 



(*+yy 



■I 



reg exp goo (r a ) 
[l+x*][l+{x + y)*] 



dx 



=c, 



i+.Uv/c) sjry 



Die Halbwertsbreite der Intensitatsverteilung I(v) 
hangt im Fall starker Absorption nach (10) und 
(7) von der Form des Schwingungsspektrums des 
Absorbers ab. Im Fall „schwacher Absorption" ist 
die Halbwertsbreite der Intensitatsverteilung unab- 
hangig von der Form der Schwingungsspektren der 
Absorber- und der Praparatsubstanzen. Dagegen 
wird die Konstante C 2 in (11), die das Verhaltnis 
der resonanzabsorbierten zu den nicht-resonanten 
Quanten bestimmt, erheblich durch die Form der 
Schwingungsspektren beeinfluBt. 



Mit abnehmender Temperatur wachst die „riick- 
stoBfreie" scharfe Linie auf Kosten der „thermi- 
schen" Linie n . Beim Ubergang zur Temperatur 
T = erreicht die „riicksto6freie" Linie ihre maxi- 
male Hohe, doch existiert daneben immer noch eine 
breite thermische Verteilung, weil die Quanten bei 
^ = zwar keine Phononen mehr absorbieren kon- 
nen, eine Phononenemission aber immer nodi mog- 
lich ist, wenn audi nur mit einer kleinen Wahr- 
scheinlidikeit. 

Fiir die „riickstoBfreien" Linien gilt bei T = 
fiir ein DEBYEsches Sdiwingungsspektrum : 

(12) 



(8) W e (E) - W % {E) - * x p*~ (0 > 



(£-£: )*+r 2 /4 
_^ exp[-(3/2)(M2mc 2 )/fc0 ] 
(£-£ ) a + rV4 
Das in der vorliegenden Arbeit besdiriebene Ver- 
fahren zur Messung der Lebenszeiten von Kern- 
niveaus ist nur bei energetisch niedrigen Kernzu- 
standen anwendbar, namlich bei solchen Kern- 
niveaus, bei denen die RiickstoUenergie E 2 /2 m c 2 
hodistens etwa das Doppelte der Abschneideenergie 
k & des DEBYEschen Schwingungsspektrums betragt. 
Nur unter dieser Bedingung tritt nach (12) und 
(6) wirklidi eine starke „riickstoBfreie u Linie aus 
dem Untergrund der „thermischen" Linie hervor. 
Der anschauliche Grund hierfiir ist, daB bei Riick- 
stoBenergien von der GroBenordnung k die Auf- 
nahme der RiickstoBenergie vorzugsweise durch 
Oszillatoren im energiearmen Bereich des Sdiwin- 
gungsspektrums erfolgen muB. Dieser Bereidi be- 
sitzt aber nur eine geringe Oszillatordichte 12 und 
die Wahrsdieinlichkeit fiir die Aufnahme des Riick- 
stoBimpulses ist entsprediend klein, was das Auf- 
treten einer starken „riickstoBfreien" Resonanzlinie 
zur Folge hat. Wenn dagegen E 2 /2 mc 2 ^> k@ , so 
konnen die im Sdiwingungsspektrum reidilidier ver- 
tretenen hochfrequenten Oszillatoren starker zur 
Energieaufnahme herangezogen werden und mit 
der Zahl der Kombinationsmoglichkeiten steigt die 
Wahrsdieinlichkeit fiir die ungehinderte Aufnahme 
des RuckstoBimpulses. 

10 In diesem Fall laBt sich ein „mittlerer" Wirkungsquer- 
schnitt a r definieren: 

/=/„ exp (-rear) mit a r = f(r/2jt) a{E) W e (E) d£ , 

6 
vgl. Anm. 5 , Gl. (9). 

11 vgl. Gl. (6). 

12 Die Dichteverteilung ist z. B. im DKBYE-Spektrum propor- 
tional dem Quadrat der Oszillatorenenergie h <o . 



134 



KERNRESONANZABSORPTION VON y-STRAHLUNG IN Ir 1 " 



215 



3. Versuchsanordnung und MeBergebnisse 

Abb. 2 zeigt die Versuchsanordnung. Der Aufbau 
der Kryostaten wurde friiher 5 beschrieben. Der Re- 
sonanzabsorber (Iridium) und ein Vergleichsabsor- 
ber (Platin) konnten wechselweise in den Strahlen- 



:tBE2M 



^\W^ 


s^> 




1 


« 


^^ 



Abb. 2. Versuchsgeometrie. A Absorber-Kryostat ; Q rotieren- 

der Kryostat mit Quelle; D Szintillationsdetektor. M ist der 

bei der Messung ausgeniitzte Teil des Rotationskreises der 

Quelle. 




Abb. 3. Relatives Intensitatsverhaltnis (/ I r — 7 P ')// Pt der hin- 
ter Iridium- bzw. Platinabsorbem gemessenen y-Strahlung als 
Funktion der Geschwindigkeit der Quelle relativ zu den Ab- 
sorbern. E=(v/c)-E ist die Energieverschiebung der 129 keV- 
Quanten relativ zu den ruhenden Absorbern. Als Strahlungs- 
quelle diente eine 65 mCurie starke Osmiumquelle, deren 
Zerfallsspektrum die 129 keV-Linie in Ir m enthalt. 

gang gebracht werden. Die Absorber und die Quelle 
befanden sich auf der Temperatur des fliissigen 2 • 
Das Szintillationsspektrometer wurde durch eine 
Photozelle so gesteuert, daB nur solche Quanten 
registriert wurden, die von der Strahlenquelle wah- 
rend ihres Aufenthaltes langs des markierten Teiles 
ihres Rotationskreises emittiert wurden. 

Abb. 3 zeigt die MeBergebnisse. Jeder einzelne 
MeBpunkt wurde aus etwa je 10 Messungen der 



Strahlungsintensitaten hinter den beiden Absorbern 
bestimmt. Die gesamte MeBzeit betrug 14 Tage. Die 
Strahlungsintensitat hinter dem Vergleichsabsorber 
(Platin) war innerhalb der Grenzen der MeB- 
genauigkeit unabhangig von der Relativgeschwin- 
digkeit v . Die eingetragenen mittleren Fehler wur- 
den aus den Schwankungen der Einzelmessungen 
bestimmt und sind immer groBer als die statistischen 
Fehler. Beim vorliegenden Experiment wurde der 
Fall „schwacher Absorption" [Gl. (II)] noch nicht 
verwirklicht. Die eingetragene, den MeBwerten ange- 
paBte Kurve wurde auf numerischem Wege nach 
(10) berechnet 13 und entspricht einer Niveaubreite 
r-(4,6±0,6)-10- 6 eV fiir das 129 keV-Niveau 
in Ir 191 . Bei der Berechnung des Absorptionsquer- 
schnittes nach (9) wurde das Schwingungsspektrum 
des Absorbers durch ein DEBYEsches Spektrum mit 
einer DEBYE-Temperatur = 285 °K angenahert. 
Diese Naherung ergibt eine zusatzliche, unter den 
Bedingungen des vorliegenden Experimentcs aller- 
dings unerhebhche Unsicherheit in der Bestimmung 
von r. 

4. Diskussion 

In der friiheren Arbeit 5 wurde fiir die partielle 
Lebenszeit r Y fiir Strahlungsemission des 129 keV- 
Niveaus in Ir 191 ein Wert von 

T y =(3,6:^)-10- 10 sec 

gef unden. Mit dem Konversionskoef fizienten a = 2,47 
nach Davis und Mitarb. 16 folgt daraus fiir die Le- 
benszeit x = r y /(l + a) = 1,0 !o'.a) • 10 -10 sec. 
Die Unscharferelation ergibt mit der im vorliegen- 
den Experiment bestimmten Niveaubreite fiir die 
Lebenszeit des 129 keV-Niveaus in Ir 191 : 
T =(l,4:S:?)-10- 10 sec. 

Wir sehen den im vorliegenden Experiment ge- 
wonnenen Wert als zuverlassiger an und verzichten 
auf eine Mittelung der Ergebnisse der beiden nach 
verschiedenen Methoden vorgenommenen Messun- 
gen, wegen der Unsicherheit in dem Wert des Kon- 
versionskoeffizienten a und weil sich bei der friihe- 
ren Messung systematische Fehler wesentlich schwie- 

13 Fiir die Rechnung wurden verwendet 7g = 3/2 (s. Anm. w ) ; 
/ a = 5/2 (s.Anm. 15 ); n = l,07-10 21 cm- 2 ; T q = T A = 88°K. 

14 K. Murakawa u. S. Suwa, Phys. Rev. 87, 1048 [1952]. 

16 J. W. Mihelich, M. McKeown u. M. Goldhaber, Phys. Rev. 

96, 1450 [1954]. 
18 R. H. Davis, A. S. Divatia, D. A. Lind u. R. D. Moffat, Phys. 

Rev. 103, 1801 [1956]. 



135 



216 



H.VOSHAGE UND H. HINTENBERGER 



riger ausschlieCen lassen als bei der vorliegeriden, 
mehr direkten Messung der Niveaubreite. 

Die in der vorliegenden Arbeit beschriebene Me- 
thode der Verschiebung „riickstoBfreier" j^-Linien 
auf mechanischem Wege gestattet eine unmittelbare 
Bestimmung der Niveaubreiten und damit audi der 
Lebenszeiten niedriger, angeregter Zustande von 
Kernen, die in Festkorpern gebunden sind. Die Me- 
thode eignet sich u. a. hervorragend zur Messung 
von Lebenszeiten energetisch niedriger Kernzustande 
in dem Ubergangsgebiet von 10 -10 bis 10 -11 sec, 
das mit der Methode der verzogerten Koinzidenzen 
schwer erfaBbar ist. Der groBe Vorteil dieser Me- 
thode liegt bei Messungen von Lebenszeiten der 
GroBenordnung 10 -10 sec darin, daB die erforder- 
lichen Verschiebungen der Quantenenergien nur von 
der GroBenordnung der natiirlichen Linienbreite 



sind und daher nur Relativgeschwindigkeiten der 
GroBenordnung cm/sec benotigt werden, im Gegen- 
satz zu der um GroBenordnungen hohere Geschwin- 
digkeiten erf ordernden Ultrazentrif ugenmethode 2 , 
bei der die thermisch verbreiterten Linien gegenein- 
ander verschoben werden. Das beschriebene Verfah- 
ren bietet dariiber hinaus im Energiegebiet weicher 
y-Strahlung eine einfache Moglichkeit, die Resonanz- 
streustrahlung von der Streustrahlung der Elektro- 
nenhulle, insbesondere von der RAYLEiGH-Streu- 
strahlung gleicher Wellenlange, ?u unterscheiden. 

Die Untersuchungen werden fortgesetzt. 

Es ist mir ein Anliegen, Herrn Professor H. Maieb- 
Leibnitz fur sein reges Interesse und fordernde Diskus- 
sionen herzlich zu danken. Herrn Professor K. H. Lau- 
terjung danke ich dafiir, daB er die Durchfiihrung der 
Arbeit am Max-Planck-Institut fiir medizinische For- 
schung in Heidelberg ermoglicht hat. 



136 



JANUARY 



PHYSICAL REVIEW 



VOLUME 55 



Capture of Neutrons by Atoms in a Crystal* 

Willis E. Lamb, Jr. 

Columbia University, New York, New York 

(Received November 21, 1938) 

The precise determination of the properties of nuclear resonance levels from the capture of 
slow neutrons is made difficult by the fact that most of the substances used for absorbers and 
detectors are in the solid state, so that the calculations of Bethe and Placzek for the influence 
of the Doppler effect are inapplicable, since these were based on the assumption of a perfect 
gas. In this paper, their calculations are generalized to include the effect of the lattice binding. 
Under the assumption that the crystal may be treated as a Debye continuum, it is shown that 
for sufficiently weak lattice binding, the absorption curve has the same form as it would in a 
gas, not at the temperature T of the crystal, however, but at a temperature which corresponds 
to the average energy per vibrational degree of freedom of the lattice (including zero-point 
energy). In cases of somewhat stronger lattice binding, the line form is found to be more com- 
plicated, and may even have a fine structure. Plots are given of the absorption line in several 
typical cases. An approximate formula for the cross section for self-indication is also derived. 



ACCORDING to the theory of the compound 
nucleus proposed by Bohr and by Breit and 
Wigner, 1 the cross section for the capture of a 
slow neutron with an energy near to a resonance 
level of a nucleus at rest in free space is given by 

* Publication assisted by the Ernest Kempton Adams 
Fund for Physical Research of Columbia University. 

« N. Bohr, Nature 137, 344 (1936) ; G. Breit and E. 
Wigner, Phys. Rev. 49, 519 (1936). 



an equation of the form 



o-o 



(1) 



4 (£-£o-i?) 2 +£r 2 

where ff , the cross section at resonance, varies 
inversely With v, the velocity of the neutron in 
the rest system, E is the kinetic energy of the 
neutron, and E is the energy that the neutron 



137 



CAPTURE OF NEUTRONS 



191 



would have at resonance if the atom were in- 
finitely heavy so that the compound nucleus 
would take up no recoil energy. For atoms of 
finite mass, the recoil energy R = (m/M)E must 
be included in the energy denominator. 2 (We 
assume that the mass M of the atom is much 
greater than the mass m of the neutron, and 
neglect terms of higher order in m/M.) T is the 
total half- value width of the resonance level, and 
is proportional to the rate of decay of the 
compound nucleus; in most cases this corre- 
sponds to the process of emission of a high 
energy gamma-ray. 

Actually, of course, it never happens that one 
has to do with a free atom at rest. This somewhat 
complicates the determination of the properties 
of the resonance level from slow neutron data. 
The atoms in a gas may be treated as free, but 
at finite temperatures, there is a Maxwellian 
distribution of velocities, and Eq. (1) must be 
modified, as has been done by Bethe and 
Placzek 3, 4 for this case. It is here necessary to 
change the resonance energy denominator ac- 
cording to the relative velocity of the neutron 
and atom, and to average over the Maxwellian 
distribution of velocities of the gas atoms. The 
proportionality of the cross section to 1/v is 
thereby unaltered, as this factor arises just from 
the normalization of the incident neutron Wave 
function to unit flux required by the definition 
of a cross section. The result of the averaging 
gives 

*=«#(&*), (2) 

where 

x=(E-E -R)/hT, £=r/A (3) 

and 

A = 2(RT)> (4) 

is the "Doppler" width of the level. 6 The function 



2ttU_ 00 



e-HHx-v)* 



l+y 2 



(5) 



2 As it is usually written, the capture cross section refers 
to the coordinate system in which the compound nucleus 
is at rest, so that no energy of recoil appears in the de- 
nominator of Eq. (1). 

» H. Bethe and G. Placzek, Phys. Rev. 51, 462 (1937). 

* H. Bethe, Rev. Mod. Phys. 9, 140 (1937). 

* We will measure temperatures in energy units, taking 
the Boltzmann constant to be unity. The results quoted 
here were derived on the assumption that the Doppler 
width of the level is much less than the energy E at 
resonance of the neutron. This condition is satisfied in all 
cases of practical interest, and we shall have occasion to 
assume it in our calculations also. 



becomes simple in the following limiting cases : 

(a) *»l/£ 2 , *-*l/(l+* 2 ), 

i.e., far enough from resonance, the line has its 
normal form. 



(b) 



1, ^->1/(1+* 2 ), 



i.e., when the natural breadth is much larger 
than the Doppler breadth, the line is again 
normal. 



(c) 



£«1, *<1/S 2 , iA->*ir*$e-K , * , I 



i.e., when the natural width is small compared 
to the Doppler width, the absorption line has an 
effective width strongly dependent on the 
temperature. 

The total activation induced in a thin detector 
by a beam of neutrons distributed smoothly in 
energy is proportional to the area under the 
absorption curve 



/ 



dEao^iZi x) = §7rrV , 



(6) 



independently of the temperature of the gas. 
Another quantity of experimental interest is a t , 
the cross section for self-indication 6 



■f**/f. 



«*E-W($vZ,.0) 

(W»*W-**(«/Sfl)J (7) 



where $ is the Gaussian error function. 

The above results are valid, however, only for 
free atoms. Most of the experiments, of course, 
have been performed with solid absorbers and 
detectors in which the atoms are bound in a 
crystal lattice of some sort with a characteristic 
Debye temperature of the order of room tem- 
perature, and if the chemical binding is of impor- 
tance, as we shall see is the case, it is clearly not 
permitted to apply the free atom theory of the 
Doppler broadening, as was done by Bethe, 4 to 
such cases as silver at ordinary temperatures. 

We shall want, therefore, to calculate the 
shape of the absorption line for an atom which is 
bound in a crystal lattice. We do not expect that 
the chemical binding will cause any difference in 

6 See reference 4, Eq. (520). 



138 



192 



WILLIS E. LAMB, JR 



the 1/v variation of the capture cross section. 
This has been shown analytically for the case of 
capture by bound protons, 7 but the result is much 
more generally valid, following in every case just 
from the normalization of the neutron wave 
function. The calculation will be made without 
detailed assumptions about the crystal model, 
but in using the final result, for simplicity, we 
will treat the crystal as a Debye continuum, and 
hence the results will not admit of an exact 
application to experimental cases. Nevertheless, 
the general features of the dependence of the 
absorption line on the characteristics of the 
lattice and on the temperature may be expected 
to be fairly independent of the detailed model. 
For just as in the theory of specific heat, there 
are several limiting cases in which the results 
may not depend on the model of the lattice as- 
sumed, so that any fairly smooth interpolation 
should approximate the rigorous result fairly 
closely. For example, let us consider a crystal 
lattice at the temperature absolute zero. If the 
lattice binding is sufficiently strong (as defined 
below), the absorption line will be normal in 
form, but centered about E=E , while for very 
weak binding, as for instance might be the case 
with a different substance containing the atom in 
question, he absorption line will again be 
normal in form, but centered about an energy 
E=E +R. Since in practice, this shift R is often 
of the order of T, the half-width of the absorption 
line, this change in the curve can be experi- 
mentally important, even though the recoil 
energy is numerically quite small. It might thus 
be possible to detect the effect of the chemical 
binding, especially at low temperatures, by use of 
different crystals, containing in common an 
element with a slow neutron resonance capture 
level, but in which the remaining elements do 
not appreciably capture or scatter slow neutrons 
of the resonance energy. 

In the intermediate cases, the shape of the 
absorption line is in general much more compli- 
cated. However, in the case of weak binding, as 
defined below, it will be possible to treat the 
bound atoms as if they were in a gas, not however 
with a temperature T, but at a larger tempera- 
ture corresponding to the average energy per 



vibrational degree of freedom (including zero- 
point energy) of the crystal. 

We must now ask for the probability W^/S,} ; 
{ a, \ ) for the capture of a neutron of momentum 
p by a definite lattice atom L of nuclear type A 
to form a nucleus B with emission of a gamma- 
ray of wave vector k when the crystal undergoes 
a transition from a state {a s \ to a state {/},}. 
Here the set of numbers denoted by {a,} gives 
the numbers a, of quanta (phonons) in the 
various modes 5 of oscillation in the lattice. We 
must consider that the final state is reached 
through an intermediate state in which there is 
neither neutron nor gamma-ray, but a compound 
nucleus 8 C with the lattice in a state {n,}. The 
usual dispersion theory gives, apart from a 
trivial constant factor, 

(Bfijk I H' I Cn.) (Cn. | H' \Aa,p) 



J5o-E+E(«.)-£(«.) + (*/2)r(«.) 



(8) 



where r(n„) is the total half- value width of the 
intermediate state (C, n,). 9 Because of the short 
range of nuclear forces and hence the inde- 
pendence of the motion in the crystal of the center 
of gravity and the internal degrees of freedom 
of the nucleus, the matrix elements of the 
perturbation H' which appear in the numerator 
of (8) can be factored into 



(9) 



(Cn,\H'\Aa,p) 

= {n, I exp (ip-x L /h) \ a,)M comp , 
(B0.\H f \Cn t ) 

-09. 1 exp (-ik-x L /h)\n t )M nd (k), 

where M T *&{k) and M CO mp are the matrix elements 
for radiation and compound nucleus formation, 
respectively, for a free nucleus, and, for example, 

(».|exp(*p«Xi/ft)|a.) 

is the matrix element for transfer of a momentum 



W. E. Lamb, Jr., Phys. Rev. 51, 187 (1937). 



8 We will ignore throughout the circumstance that the 
compound nucleus C is heavier than the atom A. This 
neglect is certainly valid if m<g.M, as is the case in practice, 
and may be seen to subject our results to a limitation on 
the effective width of the level analogous to that met by 
Bethe and Placzek (reference 5) for free atoms. As there, 
this limitation is of no importance experimentally. 

8 The curly brackets denoting a set of numbers will be 
dropped when it will not cause confusion. 



139 



CAPTURE OF NEUTRONS 



193 



p to the crystal through the Lth atom with ex- 
citation of the lattice from a state {a.} to a state 
\n t ). In practice, the lattice is in thermal equi- 
librium, therefore not in a definite state {a,}, 
and further, because of the high energy of the 
gamma-ray, the experiments will give only the 
total probability of capture, i.e., not 



W({0, 



«.)) 



but 



fit a. 



[«.}), (10) 



where the sum over the initial states of the lattice 
is weighted according to the Boltzmann factor 
g({a,}) for each state when the temperature of 
the lattice is T. Because of over-all conservation 
of energy, the magnitude of the wave vector k 
in (8) is a function of the final state of the lattice 
{/?,}. In all cases of importance, however, one 
may neglect a variation of k of the order of the 
zero-point energy of oscillation in the lattice, 
and perform the sum over the final states of the 
lattice by use of the completeness relations, 
finding 



W(E)=\M t > d \ 2 \M eomp \*Zg(a.) 

|(».|exp(ip.x*/»)|a.)|« 



XL 



[£-£ -E(rc.-«.)W] 2 +Kr(tt.)) 2 



(ID 



where the energy of the lattice has been ex- 
pressed in terms of the frequencies «, of the 
lattice oscillations. Thus one sees that the prob- 
ability of gamma-ray emission is proportional 
just to the probability of formation of a com- 
pound nucleus C irrespective of the state of the 
lattice, and this despite the fact that very often 
a gamma-ray may be emitted in a time short 
compared to the periods of oscillation of the 
lattice, giving the atom a recoil energy of the 
order of a hundred volts. Eq. (11) will be much 
less complicated if one may neglect the depend- 
ence of T(n,) on the state \n t ) of the lattice. This 
will be so except in the case, unimportant for our 
purposes, that the main contribution to V comes 
from the emission of slow neutrons, i.e., in case 
of a large elastic scattering cross section instead 
of a large capture cross section. If one were also 
to neglect the energy given to the lattice, the 
expression (11), from which we will now drop the 



factor | M c 

completeness relation would reduce just to 

l/l(E-E y+iT*l, 

since £«({a.}) = l. 

art 

We now turn to an evaluation of the matrix 
elements in Eq. (11). In terms of the wave 
functions 9 

iMxi, • • • , Xw) 

of the crystal, which is assumed to be periodic in 
a large volume containing N atoms whose posi- 
tions are denoted by Xi, • • • , Xn, this matrix 
element is 

({«.}|exp(*p.x L A)|{«.}) 



■/-/ 



dXidX 2 ' • •dX^n,*(XiX 2 - • >X N ) 



Xexp (*p- x L /h)+ at {xv • • x N ). (12) 

We introduce normal coordinates for the crystal 
in the usual form : 10 



Xo=X O +Ug 

1 
u<?= — LLe<u'C4<i;exp (iq-xo°A)+conj.), 

N* q j 



(13) 



where x© is the equilibrium position of the Gth 
atom, u<? its displacement from equilibrium, e<j 
is the unit polarization vector for the wave 
characterized by the propagation vector q and 
polarization j. The spectrum of eigenvibrations 
is determined by the periodic boundary condi- 
tions, and it is cut off at an upper frequency limit 
such that the number of degrees of freedom agrees 
with the number 3N belonging to the N atoms 
in the fundamental volume of the lattice. The 
single index 5 will often be used to denote the 
pair of indices (q, j). In terms of the quantities 



Q,=A.+A t * 



(14) 



where «. is the frequency belonging to the 5th 
normal mode, the Hamiltonian of the crystal 

10 See for instance, A. Sommerfeld and H. Bethe, 
Handbuch der Physik, vol. 24/2, second edition (1933), 
p. 500. 



140 



194 



WILLIS E. LAMB, JR, 



takes the form appropriate for a system of linear 
harmonic oscillators with coordinates Q s and 
momenta P, 



H=ZH„ 



(15) 



H s = %Mo>SQ, 2 +P. 2 /2M. 

The eigenvalues of this Hamiltonian are 

£. = (n.+J)A«.; n, = 0, 1,2, ••• 

and the wave functions normalized in terms of 

S,~Q.(h/2Mw.)-* 

are fn.(Q t ) = (2i)-*(».0-*r«« , *».(«0. 

where A» ( (&) is the »«th Hermite polynomial. If, 
for convenience, we take the rest position of the 
capturing atom Xl° = 0, as may be done without 
loss of generality, the matrix element (12) with 
the help of Eq. (13) takes the form 

n f <*€.*».({.) exp (ip-e$./(2M*«.iV)»)*«.($.), 

*J-co (16) 

where the product is to be extended over all 
the normal modes. Integrals of the form 

K(n„a,;q 8 ) = f &&n****i, m .(i.) 

are readily evaluated 11 by use of the generating 
formula for the Hermite polynomials. In our 
case, 



will enter the final result only linearly in sums 
like 



where is the Debye temperature of the sub- 
stance. Any sums of the form £s<Z« 4 . etc. would 
vanish as the transition to the continuum is 
made, which provides a justification for neglect 
of the higher powers of the q 8 2 in Eqs. (17). 
Consider now the expression 



W(a,) = Z 



U 8 \K(n„a 8 ;q 8 )\ 2 



- [£-£o-L.(w.-a.)Aa>.] 2 +ir 2 



(18) 



This is made difficult to evaluate only by the 
presence in the denominator of the term 
J^,(n s — a,)hw,. This suggests that it will be con- 
venient to group together the terms in the 
expression for which this quantity has the same 
value. One may accomplish this most easily by 
the introduction of a delta-function, writing 
(18) as 



/: 



dp8(p — £ (»«— a«) ho),) 
> 

XE 



n.|x( ) 



(£-£ -p) 2 +ir 2 



(19) 



Qs 2 - 



(p-e,) 2 

2Mhw,N 



and is arbitrarily small if we take the funda- 
mental volume of the crystal large enough, so 
that only terms to the first order in q 8 2 need be 
kept in K(n 8 , a,; q a ), as will be seen more clearly 
below. Then there are three possibilities : 

\K(a s ,a,;q 8 )\ 2 =(l-2a 8 q 8 2 )e-<» t , (17a) 

|X(a,-M,a,; 5a )| 2 = (a.+ l)g, 2 e-" 2 , (17b) 

\K(a,-l,a,;q s )\ 2 = a 8 q $ 2 e-^, (17c) 

as all the other K's are of higher order. The q s 2 



F. Bloch and A. Nordsieck, Phys. Rev. 52, 54 (1937). 



where for the delta-function, use is made of the 
usual representation 

1 r - 
«(*)=—/ dpe**. (20) 

2irJ_«, 

Thus one finds 

i r m r> a ««w 

W(a a )=—\ dpi dn 

XLn{|ii:(n 8 ,a,;5.)| 2 

n, s 

Xexp (-i»(n s -a a )ho,,) }. (21) 
From Eqs. (17), one finds 
I* = H \K(n„ a 8 ; q,) \ 2 exp —ifi(n 8 — a,)hu 8 

= e-"' i \\-\-q 8 ' i \_-2a 8 -\-{a 8 +\)e- i > iha ' 

+a^<"*-]}. (22) 



141 



CAPTURE OF NEUTRONS 



195 




Fig. 1. Plot of the ratio of "effective temperature" and 
real temperature of a crystal as a function of the real 
temperature measured in units of the Debye temperature 
of the substance. 



At this point it is most convenient to carry out 
the average over the values of the initial quantum 
numbers a„ since now each a, appears at most 
linearly. The result of the averaging is that each 
a, above is replaced by its average value a, at 
thermal equilibrium, where 



e U,lkT_\ 



(23) 



The product over the various oscillators 5 is 
then of the form 

n(i+x. 2 . 2 ) = i+LX.g. 2 

* « 

+LI>.Xr<z.V+--- (24) 

and if one remembers the smallness of the q, 2 , 
the series may be summed to give exp (£,X,g» 2 ), 
so that 



1= UI, = exp £?« 2 { (5»+ l)«-«"*- 

+a,e i > ihu ' — 2a l ,\, 
Thus one has 

W{E)=—\ dp d» . 

where the function g(/*) is given by 

g(M) = L<Z. 2 {(5.+l)e-W^ 

* 

+ a,e , '"*"«-l-2a,}. 



(25) 



(26) 



(27) 



The integral over p in (26) may be done at once 
by residue formation, and one has the generally 
valid final result 

PT(£) = 2/rReal f dy. 

Xexp[^(£-£o+*T/2)+g( M )]. (28) 

Naturally, it would be most difficult to evaluate 
W(E) exactly. One may however easily obtain 
simple expressions which are valid in the various 
limiting cases. The function g{y) is the cause of 
the complication, and it is possible to evaluate 
the integral (28) analytically only in cases 
where the values of n given by m#~1 do not 
play a dominant role. For /*0<d, one may expand 
in powers of /x and obtain 

|(m) = -^I3« 2 K 



V£<z. 2 (5.+§)(ko.) 2 + 



(29) 



The sums may be evaluated under the assump- 
tion of an isotropic crystal, i.e., the velocity of 
a wave is assumed to be independent of its 
direction of propagation, although not necessarily 
of its polarization ; and one finds 



iGO- 



■ipR-^Ri, 



(30) 



where R is again the recoil energy and I the 
average energy per vibrational degree of freedom 
of the crystal (including zero point energy). 
If the condition 

\Y+{Ri)^d (31) 

("weak binding") 

is met, only small values of ju in g{n) in the 
integral (28) need be considered, and one finds 

2 /•* 

W(E) =- dn cos (i(E-E -R) 
IVo 



Xexp (-§r M -M 2 -Ri) 



L J> 



(32) 



cos yx exp (— y— y 2 /£ 2 ) 

= (4/r 2 ),K£,*), 

where ^(£, x), 12 x, and £ are as defined by Eqs. 

12 Equation (32) gives Reiche's form of the ^-function. 
See Born, Optik (1933), p. 482. 



142 



196 



WILLIS E. LAMB, JR 




Fig. 2. Plot of the neutron resonance absorption curve in 
cold solid silver for an assumed value of T equal to 0/4. 
The curve one would obtain with free atoms is shown for 
comparison. The abscissa measures the distance from 
resonance in units of §r. If the lattice binding were very 
strong, the curve for the crystal would have the same 
form for the gas, except that it would be centered about 
the point shown by the arrow. 



(5) and (3), but now with an effective Doppler 
width 



A = 2(2*8)*, 



(33) 



which involves i instead of T. Thus we see that 
provided only the condition A+V>2d is met, 
the atoms in a crystal at a temperature T give 
the same absorption line as they would in a gas 
at a temperature I equal to the average energy 
per vibrational degree of freedom of the crystal. 
This quantity is well known from the theory of 
specific heats 



i«i(ii+2i«), 



where 



™-ii) T £ **{?-?*)- 



(34) 



(34j) 



and transverse waves. One has the limiting 
values 

e = T+d-0(6/T) T»0 (34a) 

«=!(0i+20 t )+r-o(r70 3 ). (34b) 

In Fig. 1, a plot is given of l(T)/T as a function 
of T/0 for the case that the various charac- 
teristic temperatures are equal. 

The other limiting case is /iO>l. Here one 
finds 

g(oo)=-2E<Z.2(5.+!) = 

-KG(T/ei)+2G(T/e t )l (35) 
where 

2R r Vx 



2R C Vx / » \ 

G(x) = — x* dt t( +| ) (36) 

T J \e'-l / 



with 



G(x)=xR/T *«1, (36a) 

(36b) 



2R 
G(x) = — x* *»1 
T 



where the indices / and / refer to the longitudinal 



For special ranges of values of E— E , one may 
obtain a good approximation to W{E) by 
splitting the range of integration in Eq. (28) at 
1x0=1, and in each range, using the appropriate 
expansion for g(n). One finds in this way that 
for very strong binding of the atoms in the 
crystal (d— >°o), 

1 

W(E)= , (37) 

(£-£o) 2 +*r 2 

i.e., the normal absorption line centered about 
£=£„. 

In general, however, a certain amount of 
numerical integration is required to find the 
shape of the line. To illustrate the possibility 
of using the general Eqs. (28) and (27) except in 
the two limiting cases of Eqs. (31) and (37), 
we give plots of a 2^-volt resonance energy 
absorption line in a substance at a temperature 
much lower than the Debye temperature of 
= 21O°K (Case of cold silver if one abstracts 
from the difference between 6i and 6 t ), for 
several assumed values of V. In each case, the 
curve for free atoms at the same temperature is 
also shown. One sees that for these cases of inter- 



143 



CAPTURE OF NEUTRONS 



197 



mediate binding, there is a rudimentary fine 
structure in the probability of capture which is 
suggestive of the neutron absorption lines that 
one would obtain from an atom harmonically 
bound, say in a molecule with energy levels 
separated by 6. (See Figs. 2 and 3.) 

The area under the general absorption curve 
(28) may be evaluated immediately, again under 
the assumption of footnote (4), and one finds 






dEW(E) ■■ 



(38) 



which, of course, agrees with the result for free 
atoms. The expression for the cross section for 
self-indication, which involves the integral of the 
square of W(E) is more complicated, but may 
be reduced to 



"I dfiexpl- 



Tm+«00+*(-m)]. (39) 



In the case T+^>6, this integral may again be 
evaluated by expanding g(n) for small ju, and the 
cross section for self-indication has the value 
corresponding to that for a gas at an effective 
temperature i instead of T. In the general case 
of arbitrary T, A and 6, however, one may 
derive an approximate formula by splitting the 
range of integration at n8= 1, 

,.= (W ^,[*(i + A)-*(±)] 

-f-exp[2g(oo)-r/0], (40) 




Fig. 3. Same as Fig. 2 except that the assumed value of 
r is now equal to 0. For values of r>40, one is already 
in the domain of applicability of Eq. (32). The precise 
value of r for silver is not well known experimentally. 

which may be used provided the result does not 
depend too much on the precise value of 6. 

This research was largely carried out during 
the summer session at Stanford University, and 
I wish to thank Professor F. Bloch for his gen- 
erous advice and hospitality. I am also greatly 
indebted to Professor G. Placzek and to Dr. A. 
Nordsieck for many valuable discussions and 
suggestions. 



1 



144 

ANNALS OF physics: 9, 194-210 (1960) 



Study of Lattice Vibrations by Resonance Absorption 
of Nuclear Gamma Rays* 

William M. Visscher 

University of California, Los Alamos Scientific Laboratory, 
Los Alamos, New Mexico 



It has recently been demonstrated by Mossbauer that in a large proportion 
of emissions or absorptions of gamma rays by nuclei bound in crystals at low 
temperature the recoil energy is taken up by the crystal as a whole (no pho- 
nons are emitted). This makes it possible to observe resonance absorption of 
nuclear gamma-rays without high-speed rotors or elevated temperatures, as 
had been done in the past. In the present paper we show that an extension of 
Mossbauer 's technique can be utilized to directly observe the frequency dis- 
tribution of lattice vibrations in the crystal. Specifically, if the emitter and 
absorber have the same favorable crystal structure, then the self -absorption 
cross section observed in a rotor experiment at low temperature will be pro- 
portional to N(S)/S, where N(S) dS is the number of phonon states in an 
energy interval dS, and S is the doppler shift in the gamma-ray energy in- 
duced by the rotor motion. Criteria are given for determining favorable 
cases. 

I. INTRODUCTION 

The observation of resonantly scattered or absorbed nuclear 7-rays has been 
made difficult in the past by the fact that the natural width of nuclear isomeric 
states is almost always small compared to the energy of the recoiling nucleus. 1 
Thus if Eq is the energy of the nuclear excited state, the spectrum of gammas 
emitted by a free nucleus at rest will be centered at energy E Q — R, where R = 
Eq/2Mc is the recoil energy of the nucleus whose mass is M. A gamma can be 
absorbed, however, by a free nucleus at rest only if the gamma energy is within 
the natural width V of E + R. The devices which had been used to observe 
resonance scattering before 1958 all supplied, in one way or another, the 2R 
difference between the energy of the emitted 7 and the energy which it needs 
to be absorbed. One way to supply the difference is to spin the source on a 
high-speed rotor, to Doppler shift the emitted gammas. A Doppler shift of 2R 
requires a linear velocity of Eo/Mc, which amounts to 2 X 10 cm/sec for a typi- 
cal case (Eq = 100 kev, A = 150). Another method involves varying the tem- 

* This work performed under the auspices of the U. S. Atomic Energy Commission. 
1 For a review of this field, see Malmfors (/ ). 

194 



145 



ABSORPTION OF GAMMAS BY CRYSTALS 195 

peratures of the source and absorber (or scatterer) over rather wide ranges, to 
Doppler broaden the gamma spectra. Temperature ranges comparable to the 
recoil energy must be used in order to observe the change in overlap of the emis- 
sion and absorption spectra. A third method utilizes as a source a nucleus which 
is recoiling from a previous decay. The decay product is required by coincidence 
techniques to be travelling, in a certain direction with respect to the direction of 
the subsequently emitted gamma, thus fixing the component of velocity of the 
nucleus along that direction and the Doppler shift in the gamma energy. 

A remarkable new method was devised last year by Mossbauer (#), and was 
used to measure the width of the 129-kev level in Ir 191 . His method is different 
in principle from those preceding it, in that the emitting (or absorbing) nucleus 
is not allowed to recoil, thus obviating the need to have the emitting and absorb- 
ing nuclei in rapid motion relative to one another. This is accomplished simply 
by having the nuclei bound in crystals at low temperature. Mossbauer showed 
that when the 129-kev gamma is emitted or absorbed by such a nucleus, the 
recoil momentum is taken up a large part of the time by the whole crystal, with 
no energy transferred to internal excitations of the lattice. The kinetic energy 
associated with the crystal recoiling as a whole is negligible compared to r, 
even for crystals as small as a fraction of a cubic micron in volume. Thus, the 
recoil shift discussed above is essentially zero, and many of the gammas emitted 
by a nucleus in a crystal at rest can be resonantly absorbed or scattered by an- 
other nucleus in another crystal at rest, in marked contrast to the situation 
when gaseous or room-temperature crystalline sources and absorbers are used. 

The emissions which take place without energy transfer to the lattice give rise 
to a pip of width T in the energy spectrum of gammas emitted by the crystal. 
The height of this "no-recoil pip" is a strongly decreasing function of tempera- 
ture. The absorption cross section, too, contains a no-recoil pip with similar 
properties. In both cases the pip is centered at E = E Q , the nuclear excitation 
energy. 

We will now give a brief resume of some of Mossbauer's experimental results 
and of the theory due to Lamb (8) which explains them. 

II. MOSSBAUER'S EXPERIMENTS 

In the interest of brevity, we will only outline the essential features of two 
of Mossbauer's observations. 

Mossbauer measured the transmission of Ir 191 129-kev gamma rays through a 
crystalline natural iridium (38.5% Ir 191 ) absorber. His source was Os 191 , which 
i8-decays with a 16-day half-life to a long-lived (5.6-sec) state of Ir 191 at 171 kev. 
A 42-kev y is emitted, and the iridium nucleus is left in its first excited state 
(129 kev) which has a lifetime of 1.4 X 10~ 10 sec. 

The most startling result which Mossbauer obtained was the "turntable 



146 

196 VISSCHER 





-4 




o 


4 


8 SOURCE SPEED (cm /sec) 




0.4 


-20 


-10 





10 


r 20 


30 DOPPLERl 


SHIFT (uV) 


~7 










i 


i 






> 


'\ / 






■ 


0.8 
i •> 












i i i i i 



Fig. 1. The relative intensity of 129-kev gamma rays transmitted by the iridium absorber 
as measured by Mossbauer with both source and absorber at 88 °K, as a function of source 
speed. The ordinate is dependent upon the source thickness. The width of the dip in the 
transmission for a thin absorber is dependent only on the natural width of the 129-kev nu- 
clear level. The curve here is theoretical and will be discussed in Part IV; the experimental 
points are taken from the third of Mossbauer's papers (2), Fig. 3. 

effect." Here he kept both source and absorber at 88° K, but had the source 
mounted on a turntable, so that the relative velocity of the source and absorber 
during the time the source was seen by the absorber could be controlled. Figure 
1 shows the variation of transmission he observed as a function of turntable 
speed. The absorption peak is centered at zero relative speed of the source with 
respect to the absorber; it has a half -width of about 1 cm/sec. 

When the abscissa is converted to energy units corresponding to the Doppler 
shift in the gamma energy AE = (v/c)Eo , the points can be fitted within statis- 
tical error by a Breit-Wigner curve of width (9.2 =fc 1.2) X 10" ev. This is 
interpreted to be twice the natural width of the 129-kev level, the factor of 2 
arising because the observed absorption is the result of folding an emission 
spectrum together with an absorption cross section, each of which have a "no- 
recoil pip" of width T. In the thin-absorber approximation, the absorption is 
proportional to the product of the emission spectrum with the absorption cross 
section, integrated over all energies. At zero relative velocity the pips overlap 
perfectly; at velocities large compared to Tc/E the overlap is destroyed, and 
the absorption disappears. 

The other measurement of Mossbauer's which we wish to discuss here is his 
observation of the "temperature effect." Here he had both source and absorber 
at rest, with the absorber at a fixed temperature of 88° K, and the source tem- 
perature variable from 88° to above room temperature. He measured the trans- 



ABSORPTION OF GAMMAS BY CRYSTALS 



147 
197 




400 



SOURCE TEMPERATURE °K 



Fig. 2. The effective absorption cross section per Ir 191 nucleus, for the absorber crystal 
at 88°K and the source temperature given by the abscissa. The curve is a theoretical one to 
be discussed in Part IV; the experimental points are taken from Mossbauer's first paper (2), 
Fig. 8b. 

mission of the 129-kev gammas through the absorber as a function of source 
temperature. His results, which were expressed in terms of effective absorption 
cross sections, are shown in Fig. 2. The rise in the cross section with decreasing 
temperature (quite contrary to its behavior with gaseous sources and absorbers) 
is interpreted as being caused by the increase in the probability of no-recoil 
emission by the nuclei in the source as the temperature is lowered. The absorber 
temperature was not varied in this measurement because any temperature de- 
pendence of the non-nuclear absorption cross section of the atoms in the crystal 
(mostly K-photoeffect) would obscure the temperature dependence of the nu- 
clear absorption which Mossbauer sought to measure. 

An explanation of these results was achieved by Mossbauer by modifying a 
theory due to Lamb (3), describing the resonance absorption of neutrons by 
nuclei bound in crystals, to apply to the gamma-absorption process. 

III. THEORY 

We will now explain Lamb's notation and outline his theory. 

The crystal is described by a wave function which depends on the center-of- 



148 



198 



VISSCHER 



mass coordinate of each of the AT constituent nuclei. x G = x o + u<? is the coor- 
dinate of the 6rth nucleus, where x ° is its equilibrium position and u<? is its dis- 
placement from equilibrium. If the interactions between the nuclei in the crystal 
can be approximated by harmonic forces, the crystal Hamiltonian can be written 
as the sum of 3N independent harmonic oscillator Hamiltonians. This transfor- 
mation to normal coordinates is carried out as follows. Let 

3tf I J 

Ug = £ y 2M ^ N e s [a s exp(iq s -x G °) + a* exp(-iq s -x )] (1) 

be a Fourier expansion of u<? in terms of the 3N normal modes characterized by 
polarization vectors e s , frequencies o> s , and propagation vectors q s . 
If a s and a s * satisfy 

[a s > , a s *] = 8 S ' s , (2) 

then the commutation relations between the components of u and their con- 
jugate momenta are satisfied, and if we define normal coordinates and momenta 

, (3) 

P 8 = —i A/ — 2~ {a s - a s *), 

they satisfy [P s > , Q s ] = —ihb s > s . 

The Hamiltonian becomes separable into 

3N 

H =T,H S ; 

s=l 

H s = P S 2 /2M + y 2 Mo>?Q s 2 (4) 

= fe(a s *a s + Y 2 ) = hu s (N s + K); 

where A^ s = a s *a s can, with the help of Eq. (2), be seen to have eigenvalues 0, 
1,2, • • • . The state of the crystal can therefore be specified by a set of numbers 
{ a s } which are the eigenvalues of the phonon number operator N s for each nor- 
mal mode s. To describe the entire system for a gamma-ray resonance absorption 
or emission problem we must also specify the state of internal excitation of each 
of the nuclei, and the momentum p of the gamma if any is present. Thus, for 
example, the ket | A(o; s (p) means that a gamma ray of momentum p is present, 
the crystal is in state { a s \ , and a specific nucleus is in state A . All the effects 
due to excitation of different nuclei are incoherent with one another, so it is 
legitimate to consider each nucleus separately. 

If we assume that each nucleus has two states, the ground state A and an 



149 



ABSORPTION OF GAMMAS BY CRYSTALS 199 

excited state C which has energy E Q and decays like e~ rtn , then application of 
perturbation theory yields the following results. 

The probability that a gamma of momentum p incident on the crystal in state 
{ a s } will be resonantly scattered by a particular nucleus into momentum k, leav- 
ing the crystal in state {&} is proportional to 

7 (A{(3 s }k\H'\C{n s })(C{n s ] \H'\A{a a }p) 



TT({/3.}k, {«.}p) 



(5) 



r a] E - pc + E{n s ) - E(a.) + iV/2 

where pc is the energy of the incident gamma, E(a B ) = Yl* fc(a s + %) is 
the crystal energy in the initial state, and H' is the energy operator describing 
the interaction between the gamma ray and the nucleus in question. The sum 
is over all intermediate crystal states \n s ] in which the nucleus is in its excited 
state C and no gamma is present. Energy conservation is implicit in Eq. (5); 
final states {£,} are possible only if 

E(0 S ) +ck = E(a s ) + cp. (6) 

In the derivation of Eq. (5) it is also assumed that the state of the crystal {n s } 
remains unchanged during the time the nucleus is excited. It is therefore valid 
only when the relaxation time of the crystal is long compared to the lifetime 
h/T of the nuclear excited state. 

On the other hand, if we ask only for the absorption probability, it is not 
necessary to assume that the crystal relaxation time is long. In this case one is 
led to the expression 

for the absorption probability, where T' is equal to the nuclear level width plus 
the widths of the crystal states {n s } and {a s }. The calculations will be greatly 
simplified if we assume the relaxation time to be independent of the crystal 
state; we will, in fact, assume it to be infinite, or r' = T. 

The probability for the. emission process is also proportional to (7), with {a s } 
and {n s } interchanged in the summand. 

The matrix elements of W which occur in Eqs. (5) and (7) can each be ex- 
pressed as a product of two factors. One factor depends only on the change in 
internal state of the radiating nucleus, and only weakly on the gamma energy. 
It will be hereafter omitted, since it is common to all subsequent formulas. The 
other factor describes the absorption of the gamma's momentum by the lattice. 
Thus we make the following replacements: 

(C{n s \ | H' | A{a s }p) -> [{n s } | exp(ip-x*/ft) | {«.}], 

(8) 
(A{/3.}k | H' | C{n s \) -» [{&} | exp(-ik.x L /ft) | {n.}] 



150 



200 VISSCHER 



for the absorption and emission matrix elements, respectively. We have singled 
out nucleus L at x L = x L + u L ; without loss of generality we can, following 
Lamb, choose x L = 0, the origin of our coordinate system. After substituting 
u L (Eq. 1) into the matrix elements (8) they may be factored into 3N matrix 
elements, one for each phonon state. 



ri II c /*mi n TtT I /. p-e s (q g + a*) \ 1 
[{n.} | exp(tp-u L /ft) | (a,)] = II |_». | exp ^ -^==- * S J. 



(9) 



Because of the factor 1/y/N in the exponent, no term of order higher than 
(p-e s ) 2 = Ps 2 in the expansion of the exponential on the right can give a non- 
vanishing contribution to TF({a s J> p) after the sum over [n s ] is performed and 
the limit N — ► « is taken. The terms which do contribute are 

^p \j M Zl a 4r ] « i - A- (ay- + !) + «' p '/^vf } - do) 

K L V2MM J 2MNfto) s V2MNhw s 

Since a s and a s * are phonon absorption and creation operators, respectively, we 
see that any phonon state can change its occupation number by at most one. 
Also, Eq. (10) exhibits explicitly the reason for the existence of the "no-recoil 
pip." It is that the phonon creation or absorption matrix element is of order 
1/y/N smaller than the no-phonon ({««) = \n s \) matrix element. After sum- 
ming over phonon states, the two can give contributions to W of the same order 
of magnitude. 

Of physical interest is the quantity 

W(E) = E g({a s })W(ia s },p), (11) 

where g({oc 8 }) is the probability that the crystal will initially be in a state {a s }> 
and where we have assumed, for simplicity, that the crystal is isotropic, so that 
(11) is dependent only on E = pc, and not on the crystal orientation relative 
to p. W(E) can be evaluated by methods standard to field theory. The only 
information about g({a s ] ) which is needed is the average value of the occupation 
number a s of each phonon state s. For thermal equilibrium, 2 

a s = [exp(co s /7 7 ) - l]" 1 , 

since the phonons obey Bose statistics. The result of performing the sum in 
Eq. (11) is that 

W(E) = 2/r Re f dfx expM# - E Q + zT/2) + gr(ju)], (12) 

where 

2 From now on, we will express frequencies and temperatures in energy units. 



151 

ABSORPTION OF GAMMAS BY CRYSTALS 201 

g(fi) = Z ^^- h**'*"* + (*. + D*"*"" " 2a 8 - 1]. (13) 

W(E) is proportional to the resonance absorption cross section of a nucleus in 
a crystal for a monochromatic 7-ray of energy E. 

a(E) =L aoW (E), (14) 

o (2/c + 1) x2 r 7 x x 

" = 2 (2f, + D ri r- (15) 

(To is the cross section at resonance for nuclear absorption of a gamma ray by a 
free atom, r is the total width of the nuclear excited state, while T y is the partial 
width for y-emission; I\ = (1 + «r) _1 T, where a T is the total internal conver- 
sion coefficient. If we were considering resonance scattering, the factor T 7 /r in 
Co would be squared. 

W(E) for emission differs from (12) only in that g(fi) is replaced by g(— n). 
Thus the self-absorption cross section is, for a thin absorber, 

f <r(E)W E (E) dE 3 f 

= kr <r °J W *W W *( E ) dE > < 16 ) 



<T = 



f We(E) dE 



where W E and W A are the Lamb integrals (12) for emission and absorption, 
respectively. 

If we suppose now that our emitter is" moving relative to the absorber, then 
the argument of W E in (16) should be E + v/cE = E + S, where v is the rela- 
tive velocity. It is easy to show that 



/ 



W A (E)W E {E + S) dE =^W'(S), (17) 



where W'(S) is given by Eq. (12), with E - E replaced by S, T by 2I\ and 
g(n) by g A (fx) + ^(— /*)• g* is (13) with parameters appropriate to the ab- 
sorbing crystal; g E with the parameters of the emitting crystal. Then a' becomes 

(/(#) = r*<r W'(S)/2. (18) 

IV. NUMERICAL CALCULATIONS 

If the assumption is made that the crystal is isotropic, p s 2 in Eq. (13) can be 
replaced by }^p . Then, on replacing the sum by an integral, g(n) becomes 

3 This can best be seen in Eq. (19) of Lamb's paper, where each of the p s 2 's occurs at most 
linearly. 



152 

202 VISSCHER 

^m) = # r f "" — *• [ae t>tf + (a + 1)«"*" - 2a - 1], (19) 

where N(<a) do) is the number of phonon states in dm at co, and co max is the maxi- 
mum phonon energy. The total number of states is equal to the number of de- 
grees of freedom in the crystal; 



r 



AT(co) dco = 3JV. (20) 



According to the Debye model, iV(co) = 9Nco 2 /0 3 and co max = 0. With N(<a) 
thus specified, g(p) can be evaluated quite accurately by making polynomial 
approximations to the hyperbolic function which occurs in the integrand, and 
W(E) can then be integrated on a digital computer. We have calculated <r(E) 
and o-'(aS) for the Ir 191 129-kev 7-ray with parameters taken to be E = 129 
kev, R = 46 millivolts, r = 4.6 microvolts, a K = 2.47, I c = 5/2, I A = 3/2, 
and abundance = 38.5 percent. For simplicity we assumed that the crystal 
structure of the emitter is the same as that of the absorber, and found that to 
fit the self -absorption cross section as measured by Mossbauer with both emitter 
and absorber at 88°K one must use a Debye temperature = 316°K. The cal- 
culated cross section is sensitive to the shape of the spectrum for moderate and 
high phonon energies where the Debye spectrum is known to be a poor approxi- 
mation ; therefore little significance should be attached to the difference between 
this Debye temperature and that determined from specific heat, namely 285 °K. 

The curves on Figs. 1 and 2, which respectively show Mossbauer's experimen- 
tal points for the turntable effect and temperature effect, are some of the results 
of this calculation. Our adjustment of guarantees that the turntable effect 
curve coincide with the experimental point at zero velocity, and that the tem- 
perature effect curve agree with experiment at 88°K. These two experimental 
points really represent the same measurement. 

The curve of Fig. 1, which is fitted to the bottom of the transmission dip, 
agrees satisfactorily with the rest of the points even though the calculations are 
for a thin absorber. A correction for the moderately thick (10 21 atoms/cm 2 ) ab- 
sorber which Mossbauer used would be in a direction such as to improve the fit 
for large velocities. 

The curve of Fig. 2 deviates noticeably from the experimental points at 148 
and 175°K. This deviation could probably be corrected by using a phonon spec- 
trum which is more realistic than the Debye spectrum. 

Figure 3 shows the results of the calculation for <r(E), Eq. (14), for several 
temperatures. The emission spectrum W E (E) has the same shape as <r(^) when 
it is reflected through the pip at E = E . The pip in both <r(E) and W E (E) has 
width T = 4.6 fxv. It is interesting to notice that the pip has virtually disappeared 
at T = 300°K, and that the rest of the curve has very nearly acquired the shape 



153 



ABSORPTION OF GAMMAS BY CRYSTALS 



203 







i 


A i 


\ 

100 






























160° 














292 








4 






120° 

710 , 

88° ; 






4°K 












120 °K 








/I 








10 

c 






(i 






300°K 


o 




1909 


/ 1 








— 3 




20 


/ 1 






■ 


z 


2( 


515 


/ i 








o 


4° 


/ i 


"N 






»- 
o 

UJ 

<n 

J, 2 

w 

o 

(T 
O 


r 

- PIP HEIGHT- 1 
FOR 300° 


/ i / 






- 






<ri 




VV 







.... i J 


■ i 
i 
i 

i 

i 


1 





-40 



40 



80 



120 



160 



E-E (mv) 



Fig. 3. The absorption cross section per nucleus in a crystal of natural iridium, for a 
monochromatic 7 -ray beam and a Debye spectrum for the phonons. The lowest temperature 
plotted is 4°K. 20°K would be indistinguishable from it except within 3 mv of E — E° = 0, 
where the 20° curve is up to }{ barn higher, and at the Debye energy = 27.2 mv, where 
the corners are slightly rounded. 



which one would expect for a gaseous absorber; namely, a Gaussian centered at 
E - E Q = R = 46 mv with width 2 y/RT = 69 mv. 

Some results of the calculation of cr'(S), the self -absorption cross section in 
the thin-absorber approximation, are given in Fig. 4 as a function of Doppler 
shift *S. The width of the pip in this curve is 2T = 9.2 piv. 

The variation of the self -absorption cross section cr'(O) is shown in Fig. 5 as 
a function of the Debye temperature 6, for both source and absorber at 88°K. 
The curve levels off below 6 = 200°K because here the no-recoil pip has essen- 
tially entirely disappeared. 

Figure 6 shows o-'(O) as a function of source and absorber temperature. Moss- 
bauer measured <r'(0) = 10.0 ± 0.5 barns at T = 88°K; this point has, as we 
stated above, been fit by taking 8 = 316°K. The calculation predicts that at 
liquid helium temperature the self-absorption cross section will rise to 80 barns 
and that it will decrease by only about 10% when the temperature is raised to 
20°K. 



154 



204 



VISSCHER 




S(mv) 



Fig. 4. Same as Fig. 3 except in this case the 7-ray beam is that coming from an activated 
iridium crystal at the same temperature as the absorber. The abscissa is here the Doppler 
shift in the emitted gammas induced by mechanical motion of the source relative to the ab- 
sorber. A speed of 2330 cm/sec is required for S = 10 mv. 



V. APPLICATION TO THE STUDY OF LATTICE VIBRATIONS 

The behavior of the calculated absorption cross section <r(E) (Fig. 3) at 
liquid helium temperature, with a fairly linear rise from the pip at E = E until 
E — Eq = 0, then a vertical drop, strongly suggests that the cross section for 
r < E — Eq < 6 is due mostly to absorption accompanied by emission of a 
single phonon. The bumps at higher values of E — E Q might be identified with 
2, 3, 4, • • • phonon processes. 

This conjecture can be easily expressed in analytical terms, in the limit of 
zero temperature. g(n) for T = 0(a = 0) becomes 



where 



and 



g(v) 



90 = 



9o + 0i(m), 



(21) 



'mot 



f 



NM 



do) 



155 



ABSORPTION OF GAMMAS BY CRYSTALS 



205 




Fig. 5. The variation of the self -absorption cross section as a function of Debye tempera- 
ture 0, for both source and absorber at 88°K, and no mechanical motion. JBelow about 6 = 
200°K the no-recoil pip disappears entirely for this temperature. 



01 U) = KTr / - e d °>> 

6l\ Jo Q) 



gi(fx) is just the square of the single-phonon emission matrix element, integrated 
over the phonon spectrum, and e 00 is the probability that no phonons will be 
created in the 7-emission process. The term containing exp(+z'iua>) in Eq. (19) 
is the phonon absorption matrix element ; since T = and there are no phonons 
in the initial state, it does not occur in Eq. (21). We now expand 



Mn) 



e°° 1 + 0xU) + 



toi(/i)f 

2! 



+ 



156 
206 



VISSCHER 




100 150 

TCK) 

Fig. 6. The self-absorption cross section as a function of temperature of source and ab- 
sorber. This curve has been fitted to Mossbauer's measurement of a' = 10 barns at T — 88 c 
by taking = 316°K. 



and substitute into Eq. (12), which then becomes 

W(E) = Wo(E) + Wi(E) + W 2 (E) + 
where 



9,000 /•« 

W.(B) ^ % Re / [g^)W iE ~ B ' +m) d M . 
nil Jo 



(22) 



(23) 



It is clear that W n (E) is the contribution to W(E) from n-phonon emission. 

W(E) is not directly observable but W'(S) is, because it is proportional to 
the absorption cross section <r'(S) in a thin absorber in a Mossbauer-type ex- 
periment. If, for simplicity, we again assume that the structures of the absorber 
and emitter crystals are the same, then W'(S) can be written, in the limit of 
zero temperature, 



W'(S) = W '(S) + Wi'(8) + • 



where 



Wo'(S) = 



?H 



is the no-recoil pip, and 



Wi(S) = 



2ttR 
3NT 



S 2 + r 2 

N(S) 



2y r 



(24) 
(25) 

(26) 



157 



ABSORPTION OF GAMMAS BY CRYSTALS 



is the single-phonon contribution. In general 



»••<»- SKI)"/; 



NM 



COi 



dcoi 



[ 



N(o> n ) 



207 



(27) 



doi n d(S — OJi 



.). 



If one could find a crystal which contains an isomer with parameters such that 
Wn(S) for n ^ 2 could be neglected compared to W\ for S ^ <o max , then a 
measurement of the self -absorption cross section a'(S) for r < S ^ w max would 
yield a result proportional to N(S)/S. This experiment would require a moder- 
ately high-speed rotor (capable of speeds up to 10 4 cm/sec), and ideally would 
be done with source and absorber both at absolute zero. At a finite temperature 
which still is very small compared to the Debye temperature, the measured <r'(S) 
would be proportional to N(S)/S smeared out over a range in S of the order 
of the temperature; thus any structure oiN(S) with wavelength AS < T would 
be lost. 

In order to find criteria which indicate whether or not a given nucleus and a 
given crystal could form a system with which the above experiment could profit- 
ably be performed, we will once again use the Debye model. 

For a Debye crystal g = —SR/26, and 



Wo'(S) 



Wi'(S) = 



-sine 



s 2 + r 2 ' 



67r.R0 — zr/b 



T0 3 
otherwise, 



0.^ S ^ 0, 



S 3 
6 



6 ^ SS 



se*- s -±^ $<a*», (28) 



WXS) « i (f ) e- 3 *" X I 



6 
otherwise, 



W n '(S) - \ 



n! \0 3 / T (2n - 1 



)! 



^ S ^ ; 
s < 

5 > 710. 



158 

208 VISSCHER 

^jjn-oWn(S) always converges rapidly f or ^ S ^ 0, since in this range, for 
n > 0, 

W' n+1 (S) = 6R S 2 ( 

W n '(S) " 3 2n(2n + l)(n + 1)' ^ y; 

Equation (29) attains its maximum value at S = 0, so W2/W1 is at most R/2d, 
Wz/Wi at most 72/100, etc. Therefore an ideal crystal would be one which has 
a relatively high Debye temperature, while containing nuclei having an isomeric 
state such that E 2 /2Mc 2 = R is fairly small. 

In choosing examples of nuclei which might be usable, the mechanisms by 
which the isomeric state can be excited must be considered. The first possibility 
which suggests itself is that used by Mossbauer, who used a source of Os 191 which 
/3-decays to an isomeric state of Ir . This scheme, however, involves different 
crystals in the source and absorber; thus W\ would be a superposition of phonon 
spectra for the two. If the source crystal were mostly constructed of the same 
nuclei as the absorber, with only a small doping of the parent nucleus, then this 
difficulty would be avoided, at the expense of intensity. 

Another possibility is excitation by bremsstrahlung. One might think in this 
case that one of the basic premises on which the theory rests, namely that the 
initial state in the emission process is one of thermal equilibrium, is violated. 
It can be shown, however, by using Eq. (5) rather than Eq. (7) as a starting 
point, that as long as the incoming bremsstrahlung beam contains a range of 
frequencies large compared to co max , that the same result obtains for W(E), 
provided that the relaxation time of the crystal is long compared to the isomer 
lifetime. Excitation by particle bombardment generally will leave the nucleus 
with enough recoil energy to remove it from its lattice site. 

For a system for which R/2B <<C 1, the absorption cross section per atom in 
the absorber is given, in the Debye approximation, by 

aAS ) = „ NN + I „,*->«» [_^ + 5^5] , (30) 

for ^ S ^ 0. <r NN is the non-nuclear absorption cross section per atom; pre- 
dominantly due to K-photoeffect for gammas near 100 kev, it varies from only 
a few barns to many thousand barns, depending on Z and gamma energy. 4 The 
one-phonon term in Eq. (30) can be rewritten as 

,, C v 121 /300V 21 c + 1 e~ 3Rie S v in8 , , Q1 v 

<ri (S) = —r- [ — ) or . - X 10 barns, (31) 

A \ 6 ) 21 A + 1 Ty d 

where R = 6.19# 2 M °K, if E is expressed in kilovolts, 6 and S in °K, and 
T y = fi/Yy in units of 10" sec. 

4 Tables of 7-absorption cross sections are given by Davisson (4). 



159 



ABSORPTION OF GAMMAS BY CRYSTALS 209 

cri will be large for a short-lived isomer of low excitation energy, two condi- 
tions which are hard to satisfy simultaneously. 6 As an example, if an isomer were 
found with r y = 1CT 10 sec, E = 80 kev, A = 200, = 300 °K, then R/26 = 
0.33, and <t\{8) — 2000 barns, much larger than o- NN if 80 kev is just below the 
if -absorption edge. The listed isomers seem not to have properties this favorable; 
Te 125 for example has </(0) » 5 barns, but at an energy (35 kev) where the non- 
nuclear absorption is several thousand barns. Since the background is nearly all 
absorption by bound electrons, however, a resonance scattering measurement 
would be better from the point of view of statistics and might be preferable to 
an absorption experiment. The scattering cross section is just T a /T = (1 + a T )~ l 
times the self -absorption cross section a'(S). 

VI. SUMMARY 

The technique of resonance absorption of gamma rays by nuclei bound in 
crystals at low temperature has been shown by Mossbauer to be a valuable tool 
in the measurement of natural level widths of nuclear isomers. In the present 
paper we have shown that it can also be used to determine the frequency distri- 
bution of crystal lattice vibrations, in a different and perhaps easier manner than 
the methods used previously. 6 

The assumption has been made throughout the calculations that we are deal- 
ing with simple, isotropic crystals, and the numerical estimates have all been 
based on the Debye model. The theory describing more complicated situations 
will of course be more involved, but its results should be proportionately richer 
in interpretation. If, for example, one had anisotropic crystals as source and 
absorber, a Mossbauer type experiment could separate the frequency distribu- 
tions of the normal modes with different polarizations. If the nuclei were aligned, 
still more information would be obtainable, about the spins of the nuclear states 
and possibly about the crystal structure, too. 

Numerous and fruitful discussions of these problems with many colleagues at 
Los Alamos are hereby appreciatively acknowledged. Nuclear gamma-ray reso- 
nance absorption experiments are currently in progress at this Laboratory. 

Note added in proof: Some experiments have been completed both at this laboratory 
(Craig, et al., Phys. Rev. Letters 3,221 (1959)), and at Argonne (Lee, et al., Phys. Rev. Letters 
3, 223 (1959)). The results agree with and extend those of Mossbauer to lower temperatures. 

Received: May 7, 1959 

5 A table of short-lived isomeric states is given by Alburger (5). The half -life of Au 197 , 
given there as 1.90 X 10" 10 , should be 1.90 X 10~ 9 sec. 

6 These methods involve scattering of slow neutrons or x-rays. The field of neutron inter- 
actions with solids has recently been reviewed by Kothari and Singwi (6). 



160 

210 VISSCHER 

REFERENCES 

/. K. G. Malmfors, in "Beta- and Gamma-Ray Spectroscopy," K. Siegbahn, ed., p. 521. 
Interscience, New York, 1955. 

2. R. L. Mossbauer, Z. Physik 151, 124 (1958); Naturwiss. 22, 538 (1958); and Z. Natur- 

forsch. 14a, 211 (1959). 

3. W. E. Lamb, Phys. Rev. 55, 190 (1939). 

4. C. M. Davisson, in "Beta- and Gamma-Ray Spectroscopy," K. Siegbahn, ed., p. 857. 

Interscience, New York, 1955. 

5. D. E. Alburger, in "Encyclopedia of Physics," Vol. 42, p. 83. Springer, Berlin, 1957. 

6. L. S. Kothari and K. S. Singwi, in "Solid State Physics," Seitz and Turnbull, 

Eds., Vol. 8. Academic Press, New York, 1959. 



161 



annals of physics: 9, 332-339 (1960) 

Some Simple Features of the Mossbauer Effect* 

Harry J. LiPKiNf 

Department of Physics, University of Illinois, Urbana, Illinois, and 
Argonne National Laboratory, Lemont, Illinois 

A simple description is given of the change in the state of a crystal lattice 
upon emission or absorption of a nuclear gamma ray. A sum rule is derived 
for the average energy transfer to the lattice. The probability of zero energy 
transfer is calculated. The results are general and do not assume a particular 
model for the crystal. Conclusions are presented as simple principles which 
may be useful as a guide to experimentalists. 

INTRODUCTION 

Recent experiments by Mossbauer (1) and others {2) have shown that it is 
possible for nuclei bound in crystal lattices to emit or absorb gamma radiation 
having an energy equal to that of the nuclear transition. The recoil momentum 
is taken by the crystal as a whole, with negligible energy transfer, and there is an 
appreciable probability, although small, that there is no energy transfer to or 
from the lattice vibrations. 

The theory of this Mossbauer effect is similar to that of other phenomena in- 
volving transitions in atoms or nuclei bound in lattices (3, 4) such as excitons 
or neutron capture. Calculations based on the treatments of these other cases 
have been made (5). These are rather complicated, involving the evaluation of 
difficult integrals to obtain detailed information in specific cases. Results can 
only be obtained when certain simplifying assumptions are made regarding the 
nature of the crystal. 

Because the experimental effect is small, it is of interest to look for simple quali- 
tative conclusions having a general validity (i.e., independent of the details of a 
particular model of a crystal) which can serve as a guide to experimenters in 
choosing experimental parameters. These can be obtained rather simply, avoiding 
the complications of the other treatments, if we only consider the property of 
greatest interest; namely, the change in the lattice state when a gamma ray is 
emitted or absorbed. For this calculation, the line shape of the nuclear transition 
is irrelevant, and the Breit-Wigner formula does not appear. All nuclear proper- 

* This research was supported in part by the joint program of the U. S. Atomic Energy 
Commission and the Office of Naval Research. 

t On leave from the Weizmann Institute of Science, Rehovoth, Israel. 

332 



162 



MOSSBAUER EFFECT 333 

ties drop out in the calculation except the momentum transfer in the transition 
and the nuclear mass. 

THE TRANSITION MATRIX ELEMENT 

Let us first consider the emission or absorption of a gamma ray by a free nu- 
cleus which is not bound in a lattice. This transition can be described in terms of 
a matrix element M of some operator A between the initial state i) and the final 
state /) of the nucleus: 

M = {j\A{x i , Vi ,a i )\l). (1) 

The operator A depends upon the coordinates, momenta and spins of the particles 
in the nucleus. Let us now express the operator A in terms of the center-of-mass 
coordinate of the nucleus, X, and a set of relative coordinates q which include 
spins. The dependence of A upon the center-of-mass coordinate X is determined 
completely by the requirements of translational and Galilean in variance; i.e., by 
the requirements that momentum should be conserved and that the transition 
probability for a moving observer (nonrelativistic) should not depend upon the 
velocity of the observer. For the emission of a gamma ray of momentum hK, the 
above requirements are satisfied only if the operator A has the form 

A = exp(iK -X)a(q), (2) 

where the operator a(q) depends only upon the relative variables and spins of the 
particles and has an explicit form depending upon the nature of the transition 
(electric, magnetic, dipole, quadrupole, etc.). The explicit form of a(q) is of no 
interest for our present purposes. 

Let us now consider the emission or absorption of a gamma ray by a nucleus 
bound in a crystal. The operator describing the transition is the same operator A, 
but we must now take the matrix element between initial and final states of the 
whole lattice, rather than of the free nucleus. Because the crystal forces are very 
weak compared to the internal nuclear forces, we can assume that the binding 
forces act only upon the center-of-mass motion of the nucleus and do not perturb 
the internal degrees of freedom. We can now write down an expression for the 
matrix element describing the' transition in which a gamma-ray of momentum UK 
is emitted by a nucleus whose center-of-mass coordinate is X L , while the lattice 
goes from a state specified by quantum numbers n* to a state specified by quan- 
tum numbers rif , and the internal state of the emitting nucleus changes from i) 
to/). This is 

M L = (n / |exp(zX.X L ) | n,-).(/ | a(q) \i). (3) 

The matrix element thus separates into the product of a factor depending only 
upon the lattice and a factor depending only upon the internal structure of the 



163 



334 LIPKIN 

nucleus. The transition probability is proportional to the square of the matrix 
element (3). 1 We are not interested in the absolute transition rate, but in the rela- 
tive probability of different energy transfers to the lattice. That is, we are in- 
terested in the probability P(n f , n») that the lattice will be in a particular state 
n f after the transition if it is initially in the state n,- . The probability P{n f , n») 
is proportional to the square of the matrix element (3), with the constant of pro- 
portionality chosen to make the total probability of a transition from the state 
Ui to any other state to be unity, 

T,P(n f ,m) = 1. (4) 

n f 

The dependence of the matrix element M L upon the nuclear structure matrix ele- 
ment (/ 1 a(q) \i) occurs as a common factor for all lattice states and can be 
dropped in calculating the relative probability. We therefore have 

P(n f , m) = I (n f | exp^X-X L ) | n«) | 2 . (5) 

The proportionality constant turns out to be unity, as can be verified by substi- 
tuting (5) into (4) and evaluating the sum by closure. 

A SUM RULE 

We can now derive a sum rule by making the following assumption; that the 
interatomic forces in the crystal depend only upon the positions of the atoms and 
not upon their velocities. The only term in the Hamiltonian H for the lattice 
which does not commute with X L is the kinetic energy of that nucleus, P L 2 /2M, 
where M is the mass of the nucleus and P L is the momentum. We therefore have 

[H,X L ] = -iKP L /M (6a) 

{[H,exp(iK-X L )], exp(-iK-X L )} = -(hK) 2 /M (6b) 

Writing Eq. (6b) as a matrix equation and taking the diagonal element for the 
state rii we obtain the sum rule 

D {E(n f ) - E(m)} I (n f | exp^K-Xj | m) I 2 

= Z \E(n f ) - EMMn,,*) = ^-, 

where E(n f ) and E(rii) are the energies of the states n/ and n; . 

1 We neglect the dependence of the density of final states of the emitted 7-ray upon 
the energy transfer to the lattice. This is justified since the 7-ray energy is greater than 
lattice energies by a factor of at least 10 6 . This approximation is not valid for neutron 
capture. 



164 



MOSSBAUER EFFECT 335 

The sum rule (7) says that the average energy transferred to the lattice is just 
the energy which the individual nucleus would have if it recoiled freely. 2 

Note that the Mossbauer transitions in which no energy is transferred to the 
lattice [E(n f ) = E(rii)] do not contribute to the sum rule. Thus if we want an 
appreciable probability that there be no energy transfer to the lattice the sum rule 
requires an appreciable probability for an energy transfer which is greater than 
that which a freely recoiling nucleus would receive* We will tend to get an in- 
creased Mossbauer effect when the nucleus can transfer energy to high frequency 
lattice modes; i.e., in a crystal with a high Debye temperature. If the nucleus 
under consideration is an impurity in a lattice, it should be strongly bound in a 
localized position. 

An ideal situation would occur if the nucleus were bound in a localized well and 
could oscillate in this well without appreciably perturbing the other lattice modes. 
If the excitation energy of this mode were larger than the free recoil energy 
(hK) 2 /2M), the sum rule (7) would be exhausted by a probability less than unity 
for exciting this mode. The remaining probability would all go into the Moss- 
bauer effect. 

A MORE DETAILED CALCULATION 

Let us now attempt to calculate explicitly the probability P(w,- , n») that the 
lattice remains in its initial state after the emission of the gamma ray, which 
therefore carries the full energy of the nuclear transition. We now need a more 
detailed model of the lattice. Let us assume that the interatomic forces are har- 
monic, so that a simple transformation to normal modes is possible. These modes 
will be described by normal coordinates £ s . The state of the lattice can be speci- 
fied by the set of quantum numbers {n s j describing the state of excitation of each 
normal mode. 

Let us express the coordinate X L in terms of the normal coordinates and intro- 
duce this into Eq. (5). We have 

ck'Xz, = 22 a Ls i;s , (8) 

8 

where ck is a unit vector in the direction of the vector K. We can choose the 
normalization of the normal coordinates such that 

E (<^) 2 = 1- (») 

8 

2 This result has been derived for the emission process. The author is indebted to W. 
M. Visscher for pointing out that it applies as well to absorption only if the incident 7- 
radiation has a flat spectrum over the resonance region. 



165 

336 LIPKIN 

We then have 
P({n g },{n s }) = | ({n s } | exp(iK £ a**) | {n.}) | 2 

= II | (n, | exp(z7fa L8 £ 8 ) | n 8 ) | 2 . 

8 

Elegant methods have been developed for the evaluation of expressions having 
the form (10) including averaging over all states {n 8 | with the appropriate sta- 
tistical factor corresponding to thermal equilibrium at a given temperature (6). 
We shall use a simple approximate method. We note that if we expand the ex- 
ponentials in (10), only the even powers of £ s have non vanishing matrix elements. 
We can therefore write 

P(M,M) = [lid -h s )]\ (11) 

8 

where 

h s = (n,\ (Ka Ls £s) 2 \ n s )/2 -f- terms of order K A ai s (n s \ £ 4 1 n 8 ). 
Let us expand 

II (1 - h 8 ) = 1 - E h 8 + E h s h t /2 + • • • . (12) 

We note that the expansion (12) is the same as that for exp(— E«M, except for 
the exclusion of repeated indices. If each h s is small compared to unity, even 
though E* h s may not be small, we can write 

H (1 — h s ) = exp(— E ^«) + terms of order E h s 2 . (13) 

s s s 

We therefore have 

P({w.},{n.} ) = exp E { -K 2 <*L(n s I & 1 n.) } 

(14) 
+ terms of order K 4 E als(£ s 4 ). 

s 

The expectation value of £ s 2 in the state n s is easily evaluated by noting that the 
potential energy for a harmonic oscillator }^ -Mco s 2 £ 8 2 , has an expectation value of 
(n 8 -f- %)fto) s /2, where w 8 is the oscillator frequency for the s-mode. 3 We there- 
fore obtain 

P({n s \,{ns}) « exp E {~(2n s + l)[(^) 2 /2MMfll) (15) 

s 

neglecting the higher order terms. 

3 The mass M is the mass of the nucleus emitting the 7-ray. This relation is valid even if 
the crystal consists of different types of atoms having different atomic masses. 



166 



MOSSBAUER EFFECT 337 



The factor (%K) 2 /2Mho) s is just the ratio of the free recoil energy to the energy 
of the sth lattice vibration normal mode. If the lattice is in its lowest state (at 
0°K), every n s is zero and the exponent in (15) is just the ratio of the free recoil 
energy to some average lattice vibration energy ho) Av denned by 

(/kd A v) -1 = z2 als/hus . (16) 

8 

This is a harmonic mean, in which each mode is given a statistical weight cor- 
responding to the contribution of that mode in the expansion of X L in normal 
modes (Eqs. 8, 9). We see that the probability of an effect (15) decreases very 
rapidly if the free recoil energy increases above this average lattice energy. 

The neglect of the higher order terms in (15) is justified in the region of inter- 
est, where the argument of the exponential is of the order of unity. If the number 
of relevant normal modes is large, then each individual term is small. The sum 
of the squares of these terms is therefore very small and can be neglected. 

At finite temperatures the effect is reduced because of the presence of the factor 
2n s + 1 in the exponent of Eq. (16). This reduction becomes serious at tempera- 
tures corresponding roughly to the mean lattice energy (16) where the factor 
2n 8 + 1 is different from unity in a region which contributes appreciably to the 
sum (16). 

The results (15) and (16) are general in that they apply to any crystal in 
which the forces are harmonic. The particular case of the Debye model has been 
considered by Visscher (5). We can get his result by setting a Ls = constant and 
taking a density of lattice modes which is proportional to w s 2 . For this case 

(foo Av ) De bye = %(fta> max ) = % /C0. (17a) 

r=o° 

Thus 

P({n s },K}) Deby e = exp { - y 2 (hK) 2 /2Mke } . (17b) 

r=o° 
CONCLUSIONS 

The expressions (15) and (17) lead to the same conclusions as the sum rule 
(7) ; that probability of gamma-ray emission without energy transfer to or from 
the lattice increases with the average energy of the lattice modes which are 
coupled to the recoiling nucleus and the effect becomes appreciable when this 
average energy is of the same order of magnitude as the free recoil energy. From 
the explicit form of the relations ( 15) and ( 17) with their exponential dependence 
upon the parameters of interest, we see that the effect is very sensitive to the 
experimental conditions. The experimental results (1, 2) indicate that 

P(M,{n,}) 



167 



338 LIPKIN 

is of the order of a few per cent. In this region, a change in the exponent in (15) 
or (17) by a factor of two can change the effect by an order of magnitude. 

The influence of the ambient temperature upon the effect [the factor (2n 8 -f- 1) 
in Eq. (15)] is easily understood as "stimulated emission and absorption" of 
phonons: the probability of energy exchange between the lattice and the recoiling 
nucleus increases with the degree of excitation of the lattice. 

The results might be summarized crudely by the following simple statement 
which could have been made in advance without any calculation : The Mossbauer 
experiment is described in terms of three characteristic temperatures or energies ; 
(1) the energy of the free recoil, (2) a characteristic temperature for the lattice 
(e.g., the Debye temperature), and (3) the ambient temperature. To obtain an 
effect, the lattice temperature should be of the same order as the free recoil 
energy, and the ambient temperature should be low compared to these. The ex- 
plicit results add two further important features which are of considerable in- 
terest to the experimentalist trying to increase the effect by varying parameters. 
First, there is the exponential dependence in Egs. (15) and (17), indicating that 
a small change in these parameters can have a large effect on the experiment. 
Second, there is the definition of the average lattice vibration energy (16), which 
is clearly a different kind of an average from those used by solid state physicists to 
compute properties such as specific heats. There is therefore no simple relation 
between the characteristic lattice temperature for Mossbauer scattering and the 
Debye temperature, except for very simple models which are generally not ex- 
actly valid. These two temperatures will be of the same order of magnitude, but 
the effect of the difference is very much accentuated by the exponential de- 
pendence of (15). 

It is therefore of interest for the experimentalist to look for transitions hav- 
ing a low free recoil energy, and for crystals having a high "effective Mossbauer 
temperature," Eq. (16). More complicated crystals (compounds, alloys, or im- 
purities) should be chosen for strong binding of the source or absorber atoms, 
rather than for high Debye temperatures. The latter may be due to irrelevant 
normal modes. This is particularly true in light elements where low mass, 
rather than strong binding gives high Debye temperature (7). 

The author would like to express his appreciation to F. Seitz, T. D. Schultz, 
G. Rickayzen, and R. Knox for illuminating discussions on the solid-state aspects 
of the problem; to D. R. Inglis for many stimulating discussions; to J. P. Schiffer 
and L. Meyer-Schiitzmeister for discussions of experimental results; and to R. 
M. Thaler and W. M. Visscher for information about the Los Alamos calculations. 



Received: September 25, 1959 



REFERENCES 



1. R. L. Mossbauer, Z. Physik 151, 124 (1958); Naturwiss. 45, 538 (1958); Z. Naturforsch. 
14a, 211 (1959). 



168 

MOSSBAUER EFFECT 339 

2. L. L. Lee, Jr., L. Meyer-Schutzmeister, J. P. Schiffer, and D. Vincent, Phys. Rev. 

Letters 3, 223 (1959); P. P. Craig, J. G. Dash, A. D. McGuire, D. Nagle, and R. R. 

Reiswig, Phys. Rev. Letters 3, 221 (1959). 
8. W. E. Lamb, Phys. Rev. 55, 190 (1939). 
1. R. E. Peierls, Ann. Physik 13, 905 (1932). 

5. W. M. Visscher, Annals of Physics 9, 194 (1960). 

6. L. S. Kothari and K. S. Singwi, Solid State Phys. 9, 109 (1959). 

7. J. P. Schiffer and W. Marshall (private communication). 



RESONANCE RADIATION OF 
NUCLEI BOUND IN A LATTICE 



DAVID R. INGLIS 

Argonne National Laboratory 

Lemont, Illinois 

The understanding of the processes involved in resonance radiation 
emitted and absorbed by nuclei bound in crystal lattices has acquired 
a new interest as a result of the recent revealing experiments of Moss- 
bauer 1 and others. In the literature the interpretation of these experi- 
ments has been based on analogy with a similar problem involving res- 
onance absorption of neutrons by bound nuclei, a problem treated by 
Lamb, 2 who took results from the earlier neutron resonance work of 
Breit and Wigner, and of Bethe and Placzek, who, in turn, had based the 
treatment of neutron resonances on analogy with radiation. For the 
sake of a concise, qualitative understanding of the problem it seems 
desirable to have a simple treatment of the electromagnetic radiation 
problem itself, eliminating the detour via neutrons. 

Some of the features of the problem may be presented most simply 
in terms of the analogous one-dimensional problem, and this is the ap- 
proach we shall use. We consider a nucleus as one bead of a string of 
beads, connected by springs and vibrating longitudinally. The lattice 
points are at l r = ra, where r = 1,2, ... N, and each has a longitudinal 
displacement x r , which may be Fourier -analyzed into normal coordi- 
nates q s : 

x r = (2/N)^/ 2 Z s q s sin (rS7r/N) (1) 

Here v is the number of dimensions; for the string of beads v = 1. The 
exponent v /2 is introduced for the sake of a later discussion of a crys- 
tal in which these longitudinal modes are a subset of a much larger 
number of modes, and where the coefficients for these modes accord- 
ingly become smaller (varying inversely as the square root of the num- 
ber of atoms, which in a square or cube is N"). 
The Lagrangian of the mechanical lattice is 



tWork performed under the auspices of the U.S. Atomic Energy 
Commission. 

*R. L. Mossbauer, Z. Physik, 151, 124 (1958); Naturwiss., 45, 538 (1958). 
2 W. E. Lamb, Jr., Phys. Rev., 55, 190 (1939). 

169 



170 RESONANCE RADIATION OF NUCLEI 

£ = (M/2) Zx 2 , - (K/2a) 2 (x r+1 - x r ) 2 

= (M/2)2q| - (K/a) 2q|[l - cos (tts/N)] (2) 

We consider N large, without detailed concern for the end conditions, 
as we make the following approximation in this substitution: 

I r sin 2 (rs7r/N) = N/2 Z r sin (rs;r/N) cos (rs7r/N) = (3) 

The integer s here refers to a mode of vibration of the string, indi- 
cating the number of nodes in what becomes a sine wave along the 
string as the string becomes continuous with large N; (2/N) q s is 
the amplitude of such a wave, normalized to become small with large 
N so that a mass factor M is retained in the last line of (2), even 
though a total mass NM is involved in the vibration. 

We first assume a monochromatic incident electromagnetic wave 
described by the vector potential (or by its x component) 

A(x,t) = ae i(kx - Wt) (4) 

and consider the absorption by one nucleus in the string endowed with 
an electric dipole x'e. In the real lattice, of course, each nucleus 
has a charge Ze, and the elastic force constant K results from the 
interaction of this charge with the charges of the electrons, as it 
appears in the quantum-mechanical treatment of the whole elec- 
trostatic energy of the system unperturbed by (4). Thus, the absorbing 
nucleus (which we call R) is endowed with a charge Ze, which is in 
effect at the nuclear center of mass when averaged over the time of 
lattice vibration, plus an electric dipole that shows how this nucleus 
differs from the others in interacting with the electromagnetic field 
(4). The dipole consists of a positive charge e on a proton of mass m 
and the rest of the nucleus which we call a core (though it need not be 
a closed shell), of mass (M - m). In order not to deviate from the 
total charge Ze assumed in K, the core is given an equivalent charge 
-e. 

It is then fairly obvious that the coupling with the radiation comes 
essentially through the acceleration of the proton, in the approxima- 
tion M ^> m, but let us nevertheless formulate this point and see how 
the correct position coordinate enters in the vector-potential term. 
The vector potential enters! the Lagrangian of a many-particle sys- 
tem thus: 



tThe simple proof of this consists in showing that the Lagrange 
equations in cartesian coordinates then include, among other terms, 
the deflecting force of the magnetic field, F ma g = (e/c)v x H with 
H =Vx A, as is shown for example in Van Vleck's book, "Electric and 



RESONANCE RADIATION OF NUCLEI 171 

£ = (1/2) 2 n mv^ - V + Z n (e/c)v n - A (5) 

If we neglect the interaction of A with the nuclear charges Ze and 
with the electrons, which would be of importance in problems of crys- 
tal optics, we have for our system 

£ =(1/2)M2 (N " 1) 4+(1/2)(M- m)4 + (l/2)mx£ 

- (K/2a)Z(x r +1 - x r ) 2 - V(x p - x c ) + (e/c) 

x [xpA(xp) - x c A(x c )] (6) 

The sum (N - 1) includes all N values of r except r = R. Here x p 
and x c are the positions of the proton and core, and we of course re- 
duce them to the center-of-mass and difference coordinates for the 
R-th nucleus, 

xr = [(M - m)x c + mXpJ/M x' = x p - x c (7) 

and have 

£ = £v + (l/2)/ix' 2 - V(x') + (e/c)[(x R + (/i/m)x')A(x R + (/x/m)x ; ) 

- (x R - (m/M)x')A(x R - (m/M)xO] (8) 

Here we have the usual reduced mass, n = m(M- m)/M, and the first 

term in £ is a sum over N terms, not just Z over (N - 1) 

terms excluding x„ as in Eq. (6). The two cartesian momenta are 

P R = 8£ /3x R = Mx R + f f = (e/c)[A(x R + (/i/m)xO 

- A(x R - (m/M)x')] 

p = 3 £/dk' = \xk' + g g = (e/c)[(M/m)A(x p ) - (m/M)A(x c )] 

(9) 

If, neglecting the mass of the proton compared to the mass of the 
core, we put m/M a 0, ju/m « 1, we have 

f « (e/c)[3A/ax] x x' + - g « (e/c)A(x p ) (90 

Magnetic Susceptibilities, ,, Oxford University Press, New York, 
1932, p. 19. The fact that the acceleration in the transverse direc- 
tion is involved in the longitudinal radiation process means that the 
problem is not truly one -dimensional, although the essential lattice 
part of the formulation is. 



172 RESONANCE RADIATION OF NUCLEI 

The Hamiltonian contains for these two coordinates the terms 

(P R - f) 2 /2M + (p - g) 2 /2M (10) 

and we may here first use the approximation (9') and further neglect 
f altogether, both because it appears as f/M, which is small com- 
pared to g/jLt from the mass disparity, and because the gamma- ray 
wavelength of interest is rather large compared to nuclear dimensions. 
With these approximations, and with neglect of g 2 , the Hamiltonian 
JC= 2pq- £ is 

JC = (1/2M)[2 (N_ 1] P 2 + (P N - f) 2 ] + (K/2a)Z(x r +1 - x r ) 2 

+ (p-g) 2 /2ju + V(x') 

« (1/2M) 2P 2 . + (K/2a)Z(x r +1 - x r ) 2 + p 2 /2p. + V(x') 

- (e/mc)p.A(x p ) (11) 

as some readers would have considered obvious without all this talk. 
One purpose of this detailed discussion has been to show why it is x p 
that appears as the argument of A. 

In the approximation f = 0, the lattice -point momenta are simply 
P r = Mx r , and the transformation to the normal coordinates and mo- 
menta p s and q s proceeds as in (2), 3C = 2 S 3C S + 3Cn + 3C 1 

3C S = p|/2M + (Mw|/2)Qs w s ~ (2K/Ma)(l - cos tts/N) 

p s = d£/dq s = Mq s — (ft/i)3/8q s 
JC N = p 2 /2^l + V(x') 

3C'=-(e/mc)p-A(x p ) (12) 

The lattice part of the Hamiltonian, I3C S , is the sum of terms s re- 
ferring to the normal modes separately, and so is separable into os- 
cillator equations each of which involves one factor in the composite 
wave function. The time -independent oscillator wave equation, 

[-(1i 2 /2M)8 2 /aq 2 s + (Mu)|/2)q| - E s ] U s (q s ) = (13) 

is expressed, as shown in Eq. (13'), in the usual convenient oscillator 
coordinates ? s = (wsM/h^qgi 

(-9 2 /9«s + *l -*s)U s (!s) = A s =2EsAx) s = 2n s + 1 (130 



RESONANCE RADIATION OF NUCLEI 173 

The well-known solutions U s (| s ) have between them the following 
matrix elements of £ 2 , which we shall need shortly: 

<n|| 2 |n+2> = (l/2)(n+l) 1 / 2 ( n +2) 1/2 

<n || 2 |n> = (n+1/2) 

<n || 2 |n-2>=(l/2)n 1/2 (n-l) 1/2 

<n|« |n + l>=[(n+lj/2].* 

<n|| |n-l> = (n/2) V2 (14) 

The nuclear part of the wave equation 

(H n - e n )u n (x) = (15) 

need not be specified in any detail, as all we shall demand of the nu- 
clear wave functions u n (x) is that there be an electric dipole transi- 
tion between two of them, Ui and Uf, characterized by an energy dif- 
ference ei - ef = hco . 

The interaction with the electromagnetic field (4) is of course 
treated as the perturbation, and we shall be particularly interested in 
the relative contributions made by various Fourier components that 
enter here, without concern for the absolute strength factor C. 

ac' = (ine/mc)(a/ax')A[x R + (/x/mjx'] 
w Ce i[k(x R + x') - cot] 



= Ce" icjt e ikx ' ne ibs * s 
s 

= Ce iwt (l + ikx' + -.) I V e ib s^s 
s= 1 

N ^ 2 



N 

1 + i L Ms 

s = n 



\s = n 

C = ietefi/mc b g = C^g" 17 * sin (Rstt/N) 

C 1 =(2/Nf /2 (t/M) 1/2 k (16) 

We see that the lattice Fourier components of the motion of the 
nucleus R, on which the dipole rides, enter the coefficients b s , which 

- v/2 
contain a factor N and thus in general become small as N be- 

comes large, but we shall see that they make a finite contribution be- 
cause one obtains a sum of about N^ terms in b g . 

In the fourth line of Eq. (16) the terms in the product have been 



174 RESONANCE RADIATION OF NUCLEI 

separated into two classes and only those exponentials with s ^n 
have been expanded in a power series. The others have been retained 
as exponentials to avoid the "infrared catastrophe."! For the long- 
wave lattice vibrations, s <^N, we have from Eq. (12) 

w| = (2K/Ma)(l - cos 77s/N) * 77 2 Ks 2 /N 2 Ma = (ti/2) 2 (sw n /N) 2 

b s < b s /sin (Rstt/N) = (2/N)^ /2 (li/Mu> s ) 1/2 k 

= 2 3 / 4 (2/N) 1/2(l " " X) (h/7TMs) l/2 (Ma/K) 1/4 w /c (160 

- (l/4)2 3 / 4 (2/N) 1/2(l ' " ^(meAMs) 1 ' 2 {m e c 2 /^ 2 (a/v)V» 

«2(2/N) 1/2( ^ 1) s-V2 

Here we have introduced the velocity of sound, v = (Ka/M) 1/2 , and in 
the sample evaluation following the arrow we have used the typical 
value v/a = 3x 10 13 sec -1 (the frequency corresponding to the time 
for sound to travel a lattice distance, or approximately the Debye 
frequency) and we have used "no; = m e c 2 /4, since the gamma ray of 
Ir 191 used in Mossbauer's first demonstration of the effect has the en- 
ergy 129 Kev. We have taken M = Am = 200 x 1840 m e . 

Thus, the dividing line, which we have called n in Eq. (16), between 
b s greater than and less than unity, is in the neighborhood of s « 4 
for one dimension and there exists no such distinction for more di- 
mensions (given the crystal constants and gamma energy of our ex- 
ample).! For considerably larger s one may expand the individual 
exponentials, but there are so many terms in the sum of factors in 
the product that the validity of the expansion must be further examined. § 

Formulation of the Transition Probabilities 

In the usual time -dependent wave equation 

[JC - (VDO/at)]* = (17) 

one considers the approximate solution of interest to be a superposi- 
tion of the initial state i and a growing admixture of the final state f, 



tThe infrared catastrophe does not occur for three dimensions, as 
we shall see. For that case, v = 3, one may set n = 1 and eliminate a 
part of the following discussion, which applies explicitly to the one- 
dimensional model and is presented in order to facilitate an under- 
standing of the importance of the three dimensions of actual crystals 
in determining the shape of the absorption spectrum. 

jThe dividing integer n becomes greater as the crystal binding 
becomes weaker and the limit of weak binding will be of interest. 

§See page 187. 



RESONANCE RADIATION OF NUCLEI 



175 



^/ = a i (t)^ i +a f (t)^ f 



18) 



where 



i(Ei/R)t 



^ = Ui(x')n s U ng (? s )e 



are solutions of Eq. (17) without 3C'. The coefficient ai is initially 
unity and the initial rate of growth of af is 



a f = (l/ffl)/*f ac'^i d|> ... d^ N dx' = De i(2 sK - n s )c s + u, - c)t 

= De iAa;t 
D = (Ck/fi) < f | x'| i > D^ 

ik <f |x'| i> w / u* e lkx 'ui dx' = / u*(l + ikx' + •••)u i dx' 

n-1 



D 1 = <n l * ••• n n _i 



n e 
s = l 



ib s ^s 



n i'" n n-l> 



D 2 - < n n / • • • n N 



N / N \ 

l + i £ Vs-U/2) £ b s^s 
\s = n i 



s = n 



n n --n N > 
(19) 



Here oj = (ef - ei)/h is the resonant frequency of the nucleus at rest 
without recoil, Ef = ef + 2 s n s 'a;s i s the final energy of the system 
and, similarly, E^ = ei + Z s n s oj s , and Aco is the amount by which the 
incoming wave is out of exact resonance with the energy change from 
the initial to the final condition of the system as a whole, nucleus plus 
lattice. In Eq. (19) we have selected the electric dipole term of the 
nuclear transition as the usual case, but this is not essential. The 
early values of af , for small t, are 



H 



f De iAwt dt = (D/iAo>)(e iAajt - 1) 



and thus the probability per unit time that the transition has been 
made to the final state, for t small enough that a^ remains not far 
from unity, is 



176 RESONANCE RADIATION OF NUCLEI 



(l/t)|a f p = (D/Aco) 2 |e iAwt / 2 f |e iAwt / 2 -e^^f 

■^W (20> 

The last factor, the square of the square bracket, is clearly a func- 
tion which approaches unity for very small Au>t and becomes zero 
for Aco = 2ir/t. Thus it describes a resonant peak of half -width about 
fl/t, becoming narrower as the time becomes longer, in keeping with 
the uncertainty principle, and the factor t in Eq. (20) makes the peak 
higher as the time becomes longer. For larger values of Au>t, the 
function goes through a series of small maxima of rapidly decreasing 
amplitudet which have no great importance compared to the central 
sharp peak. 

In the resonant absorption of gamma rays, the time t during which 
the radiation causing the transition takes place is taken to be essen- 
tially the half- life of the emitting state (the same as the half- life of 
the final state of the absorbing nucleus, the two nuclei being identi- 
cal). 

The shape of the resonant peak given by Eq. (20) is similar to, but 
not the same as, the shape factor of the dispersion formula, J 
|(Ao>) - ir/2|* 2 . The latter is derived by treatment of the steady state 
resulting from the equilibrium between the absorption here treated 
and the corresponding emission that is responsible for the half- life 
of the final state f ; the nonresonant "tail" obtained does not display 
the "beats" found in Eq. (20) for large Aw. To obtain a qualitative 
understanding, it does not matter which resonant line shape is used. 

As the half-life in a case of interest is known from experiment, 
we consider t given. The structure of the theory is then as follows: 
In exploring the shape of the absorption spectrum, we consider at 
first a given o>. There is then a range of energies of cu within about 
Aw = 2u/t, in which some final states f , n s r may be found for which 
the matrix elements in D of Eq. (19) do not vanish. One could think 
of searching out these states, and then computing their contributions 
by means of Eq. (20), and adding the values obtained to get the absorp- 
tion intensity at oj. 

Alternatively and more conveniently, we may first consider all the 
final states to which matrix elements exist, and calculate for each 
state, by means of (20), the contribution it can make to the absorption 



tThe shape of this function, as well as the formulation of this sim- 
ple (Dirac) approach to the radiation problem, is to be found conven- 
iently in L. I. Schiff, "Quantum Mechanics," McGraw-Hill, New York, 
1949, pp. 189-193. 

JG. Breit and E. Wigner, Phys. Rev., 49, 519 (1936). 



RESONANCE RADIATION OF NUCLEI 



177 



at various a>'s within the range Aa; of its resonant frequency 
u) + 2(n s i - n s )a> s , as its influence is spread out by the line shape. 
The various contributions at each a? must then be added to obtain 
the absorption spectrum. 

To do this we must examine the lattice matrix elements. When the 
constant and nuclear elements which are the same for all terms are 
left out, the lattice factor for the low frequencies, D x of Eq. (19) is a 
product of (n- 1) factors of the type 



<n £ 



iM 



sss 



ib s l 



*S>=/U* tt)U n ({)e 1D ** d{ 



(20') 



-£ 2 /2 
where U n (|) = H n (£) e ' . The hermite polynomial H n is a sum of 

about n/2 powers of £ in terms of alternating sign, giving about n/2 
nodes on each side of zero, and about n/2 maxima of H n with ampli- 
tudes that increase until the wave function is cut off by the exponential 
factor. The appearance of Uf is shown in Fig. 1, along with dotted 
lines giving the classical probability distribution for a harmonic oscil- 
lator. The faithfulness to the correspondence principle is clear, and 
in the central region, where the parabolic potential curve is relatively 
flat, U n (£) is given approximately by the standing- wave wave function 



of a free particle in a box, U n « e 



iA 1 ^ 



iA l/2 | 



with A= 2n s + 1, 




Fig. 1. Square of the oscillator wave function for n s = 10. The 
broken lines show the probability distribution for a clas- 
sical oscillator which slows down and stops at the end 
of its swing. 



178 RESONANCE RADIATION OF NUCLEI 

satisfying Eq. (13') with £= 0. In the matrix elements (20'), with 
b s ^>l, the exponential factor e^° s * varies rapidly in phase and 
makes the integral very nearly vanish if the product of the two wave 
functions varies much more slowly, as it does if n s /»n s . For the in- 
tegral (20') to have a value nearly unity, it is necessary for the prod- 
uct of the wave functions in the important central region to have the 

same periodicity as e s . If n s = 0, as it would be at a very low 
temperature, Uj varies slowly and has no periodicity; the periodicity, 
therefore, comes from Uf and we have A 1/2 = (2n s > + 1) 1/2 = ±b s if 

e s is to keep the same phase through the important range 

of integration and if the matrix element is thus to be large. With 
v = 1 and with b s taken as an average over several adjacent modes 
s (because the radiating nucleus R is in a position to have good "lev- 
erage" in interacting with some of the normal modes of vibration and 
not with others), we may use (sin 2 ) av = 1/2 in Eq. (16) and then have 

(2n' s - l)w s =fik 2 /NM 

The contribution to Aw of Eq. (19) by all terms of this type is 

n-1 

£ nJ3W S ~nfik 2 /2NM 
s= 1 

This is the recoil Doppler shift produced by the motion of the radiating 
atom in phase with other atoms in the same region in these long-wave 
vibrations. A large block consisting of about N/n atoms can move to- 
gether without exciting too much photon energy and can recoil as a 
unit. In the limit of very weak binding, all the b g values become 
large, the sum extends to N, and we obtain the usual recoil Doppler 
shift for a free atom initially at rest, 6oo = Hl^/^M. 

If n s and n' s are both large, we may use the free-particle stand- 
ing-wave approximation for both and have (2n^ + 1) - (2n s + 1) = ±b s . 
The contribution to Au> of a given normal- mode excitation n' s is 

K -n s )co s = (nfcVi + nV 2 )(n^/ 2 _ n ^) Ws «(2n^ ± b s /2 1 / 2 )( ± b s /2 1 / 2 ) 

Addition of these contributions from the various modes s would in- 
volve a combinatorial problem, such as taking account of the fact that 
there are relatively many states with very small total contribution 
because the ± sign is about equally divided between plus and minus. 
This would amount to a treatment, by means of fourier analysis, of 
the ordinary Doppler shift that results from the thermal motion of 
that large block of atoms whose common motion may be described by 
these long-wave modes. One sees that the same large block is again 



RESONANCE RADIATION OF NUCLEI 179 

involved by noting that the statistical problem may give a factor n 1/2 
from summing over n terms with random sign, and one thus has a 
factor (n/NM) where a factor (1/M) would appear in the Doppler shift 
of a free atom. 

In favorable cases, n is less than N by several orders of magni- 
tude, a fact which both keeps the spike sharp and which means that the 
great majority of normal modes fall in the category of s greater 
than n. 

Now let us consider the high-frequency end of the phonon spectrum. 
We do so with the idealized approximation that the b s values are suf- 
ficiently small for us to use the expansion in D 2 of Eq. (19). There 
we have made the expansion with the summation in the exponent, and 
require the sum to be small. We could take instead a product of ex- 
ponentials for the individual modes s, expand each, and require only 
that the individual term b s £ s be small, a seemingly less stringent 
requirement. Even in this case the expansion would be questionable 
as s approaches the borderline n. We must thus leave some region 
untreated, and consider the lower limit n somewhat larger than that 
borderline n. After expanding the individual exponentials, one has a 
product of (N- n) sums and, in any order above the first, one has an 
enormous number of products which are not obviously negligible and 
which yet do not appear when one expands as in (16). It will be shown 
below that the two forms are equivalent, and thus the less stringent 
requirement is sufficient. 

With this limitation on the range s under consideration, the short- 
wave factor obtained by taking the square of D 2 in Eq. (19) is 

|D 2 | 2 ~|<n n -Dfc|l+ i2 s b s | s - (l/2)2 s b||||n n - n N >| 2 

U-(l/2)2 s b|<n s || 2 s |n s >] 2 if all npnj 

k| |<n s ± 1 || s |n s > | 2 if all nj=m except ng=n s ±l 

{l/4)b||<n s ±2 ||||n s > | 2 if all nj=nj except ng = n s ±2 

(21) 

For a given initial state n x ,,, n N there is just one final state of the 
first type, and its lattice factor may be written more explicitly, 

|D 2 | 2 ~{l-I s [n s + (l/2)]b 2 s +".} if all nj = nj (210 

1 
There are about N states with lattice factors of the following type, 

one for each of the lattice modes s that may be excited: 

|D 2 | 2 ~(l/2)(n s + l)b| if all nj = nj except ng + 1 (21") 



180 RESONANCE RADIATION OF NUCLEI 



As long as most of the nj are not zero there are about N states with 
factors of the type 

|D 2 | 2 ~ (l/2)n s b| if all nj = nj except ng = n s - 1 (21'") 

If we confine our attention to powers of b s no higher than b|, these 
are all the transitions we need to consider. Transitions to the single 
state of the type (21') give an absorption line having the natural width 
at the undisplaced resonant frequency u> , except as slightly broad- 
ened and displaced by the long-wave modes in D 1 . The second type 
(21 '0 gives the possibility that any one of the lattice modes may ab- 
sorb one quantum of excitation, contributing to the spectrum in the 
immediate neighborhood of (oj + w s )- Since there are many lattice 
modes and thus many u> s , these modes, through this term, can con- 
tribute a broad part of the spectrum. The last type (21'0 corresponds 
to the nuclear excitation receiving most of its energy from the gamma 
ray, but part of it from the crystal mode s, and the factor n s as- 
sures that this cannot happen unless mode s is excited. This process, 
of course, contributes to the spectrum of the low -energy side of co . 
In the single term of type (210 there is subtracted a sum of n terms 
in b s . This compensates in the total intensity the absorption by the N 
states of type (21") and of type (21'"), in this order, leaving the total 
intensity unaffected by the influence of the lattice. 

It is clear that the shape of the absorption spectrum depends on 
the prevailing magnitude of the n s , that is, on the degree of excita- 
tion of the crystal modes, and this, of course, depends on the temper- 
ature. At absolute zero, with all values of n s = 0, the "central spike," 
or almost undisplaced line, at u) is considerably weakened by a wing 
covering the range of crystal-mode energies on the high-energy side. 
At moderately low temperatures, n s will be fairly large for the low- 
frequency modes with small s, and n s will be small for large s, as 
s approaches N. The low-frequency modes will make their very 
small contribution to the Doppler broadening of the spike, and the 
high-frequency modes (as well as the untreated intermediate modes), 
will start to develop a wing on the low-frequency (phonon de-excita- 
tion) side, which further detracts from the intensity of the spike. As 
the temperature becomes higher, the central spike disappears and the 
strong outer part of the wings is also much enhanced. The fact that 
according to Eq. (22) the spike is even negative (and not only in the 
limit of large n s ) shows that the simple approximation used is not 
valid, particularly in its neglect of terms of order bj. An improved 
approximation is discussed below. 

As a simple extreme case, let us consider further the shape of the 
spectrum in the limit of very low temperatures where few phonons 
are present. Consider in particular the shape of the high- energy wing 
given by (21") with all n s = 0. The energies of the lattice modes are 



RESONANCE RADIATION OF NUCLEI 



181 



fujO s with u>s = (l/2)w|^(l - costts/N) and cu|g = 4K/Ma, according to 
Eq. (12). This type of frequency spectrum as a function of a> s is il- 
lustrated vertically in Fig. 2, for the case N = 20. As a function of 
u> s , or oscillator energy, it is plotted horizontally, the transfer being 




Fig. 2. Distribution of frequencies of the normal modes of a 

string of twenty beads (N = 20), showing graphically how 
they bunch at the high-frequency end, in terms of the 
first line of Eq. (12). The lower graph gives the corre- 
sponding density of states for a larger value of N. 



made by means of the parabola shown. For the large values of N that 
are of interest, the lines crowd together in the extreme co s - u>n> to ° 
closely to permit drawing them. For a very large N, the spectrum is 
almost a continuum of varying density and the number of modes per 
unit io is 



182 RESONANCE RADIATION OF NUCLEI 

N(co s ) = (awg/as)" 1 = (2 3 / 2 N/nco N )(l - cos 77s/N) l/2 /sin (tts/N) 

= (2N/ff« N )/[l - (co s /co N ) 2 ] 1/2 (22) 

Here we have used the relation sin 7rs/N = 2(co s /con)[1 - (cos/con) 2 ] / 
in transforming from a function of s to a function of co s . This is 
plotted to show the comparison with the horizontal spectrum in the 
lower part of Fig. 2. 

In practice the natural line width w of the gamma line will be 
greater than the spacing of the levels, particularly at the crowded 
upper end of the spectrum, because N is very large, and the intensity 
of absorption is given by the integral of (21 ") multiplied by (22), or of 
b|N(co s ) -WsVtl - (co s /con) 2 ] , multiplied further by the line shape 
and integrated over the entire width of the line. For qualitative pur- 
poses it suffices to square off the line shape and let it have a constant 
height over the width w (on the scale of co s ). The number of states 
involved, or the integral of (22) is then indicated by the shaded areas 
given as examples in the lower part of Fig. 2, and these are to be 
multiplied by the median Wg 1 to obtain the relative intensity of absorp- 
tion (since, again with an average sin 2 = 1/2, we have b| = (l/2)Cf/co s , 
and C\ is a common factor). 

The line width thus blunts the singular peak of the function N(co s ) 
at the upper limit of the spectrum, con- The area of the shaded region 
under this peak is 

co N w/co N 

/ dw/[l - (co/co N ) 2 ] V2 « co N ! oV(2e) 1/2 = (2co N w) 1/2 (23) 
con~ w 

The intensity at the peak is thus approximately given by (2w) 1/2 wjJ 
The corresponding intensity, that is, the intensity with the common 
factor (NC?/7tcon) again omitted from /b|N(co s N(co s ) dco s , in the non- 

w 

singular part of the spectrum is w/co s [l - (co s /con) 2 ] and the ratio 
of this to the intensity at the peak is 

Kco s )/I(co N ) = (w/2w N ) 1/2 ( WN /a; s )/[l - (u> s /w N ) 2 ] 1/2 (24) 

Here we have considered w<^Ccon> and nave called the peak intensity 
I(con)> although in this treatment the peak is reached more exactly at 
con ~ U/2)w and the intensity falls to zero again at con + (1/2 )w. 
These points are used in the sketch of the ratio (24) shown in Fig. 3 
for the case in which w = (1/10)con is rather large. For a smaller w 
the peak is obviously narrower and higher, relative to the rest of the 
spectrum. The peculiar shape of the peak is artificial, arising from 



RESONANCE RADIATION OF NUCLEI 



183 




Fig. 3. Rough sketch of absorption spectrum of a gamma ray 

with assumed square intrinsic line shape having a width 
w = wn/10) of a nucleus in a one- dimensional lattice. The 
broken curve indicates that the rise toward a cusp at the 
undisplaced line is steeper at finite temperatures than at 
T = 0. 

our use of a rectangular line shape. With the realistic line shape given 
by Eq. (20) it would of course be rounded off nicely. 

For small cj s , much less than co^, the spectrum is linear in 
u>n/ws' This ma Y still apply in the low end of the high-frequency re- 
gion n < s « N. When we introduce a small but finite temperature, 
T, with kT<fiu>N, there is a region of small oj s with 'ha; s <kT 
wherein there is a Boltzmann distribution of excitation of the phonon 
quantum numbers n s , and, for our qualitative discussion, we may in- 
sert the average excitation n s = kT/nco s . Equation (21") with n s = 
was used as a factor in obtaining (24), and in the region of (24) where 
w s <a)jj, reintroduction of n s has the effect of multiplying by n s + 1 
«n s . For small u> s Eq. (24) then becomes 

Kw s )/I(w N ) * (kT/Ha4)(wa> N /2) 1/2 



This gives a steep approach to a singularity at cos = and for 
w = (l/10)w£j, and kT/hu; N = 1/2 suggests a curve about like that 
shown by the dotted line in Fig. 2a, the peak at s = N being practically 
unaffected. In the region s«n we expect (21") and thus also this dot- 
ted curve to be an upper limit. For a higher temperature, kT>nu>N, 



184 RESONANCE RADIATION OF NUCLEI 

(23) is also to be multiplied by a factor n^-kT/ftoJN and thus (24) is 
multiplied by u; N /a> s , giving 

It(w S )At(wn) = (w/2co N ) 1/2 (co N /u> s ) 2 /[l- (u> s /u N ) 2 ] ^ (25) 

for (jo s < (o>n~ w )* Tnis curve is similar to the previous one indicated 
by the dotted line but with the low frequencies further enhanced. 

This gives us an idea of the shape of the absorption spectrum for 
the one-dimensional problem on the high-energy side, gamma-ray en- 
ergy having been absorbed by the lattice vibrations as well as by the 
nucleus. In the last step we have considered the temperature high 
enough for us to consider n + 1 «n s . In this approximation the emis- 
sion strength (21 '") is the same as the absorption strength (21"), and 
we have on the low -energy side a distribution of approximately the 
same shape, part of the energy for the nuclear transition being sup- 
plied by the lattice vibrations. Thus, we expect the spectrum to con- 
sist of two wings, each with something like a peak near its outer ex- 
tremity, and both flanking a central ' 'spike. " The peak at the high- 
frequency end of each wing is probably not to be expected to be so 
sharp in nature, because it comes at the frequency where one cuts 
off the idealized "Debye" spectrum of the crystal, and the actual cut- 
off at high frequencies is doubtless somewhat different, depending on 
details of the crystal structure. 

As co s decreases below about (1/2)o>n, the intensity rises seem- 
ingly toward a cusp at the center, but the approximation becomes un- 
reliable as the cusp is approached. If we take w/w^ < 0.07, let us say 
w/ojn = 1/20 rather than 1/10 as above, then the line width would 
pick out individual lines in Fig. 2 for small values of w s , and for each 
of them would contribute a peak to the absorption spectrum, with a 
small energy shift corresponding to a one-phonon jump at that fre- 
quency co s . If w spans the undisplaced position w s = on the scale 
of Fig. 2, it gives rise to the undisplaced peak in the spectrum corre- 
sponding to no phonon jump for any s. For this no-phonon transition, 
the intensity at zero temperature is given by the term unity from Eq. 
(210 for all s, rather than to have (l/2)b| from Eq. (21") for one of 
the s values. Thus, the intensity of the peak of the wing relative to 
that of the central peak [according to both Eq. (23) and the remark 
following it] is 



[(l/2)(2w/co N ) 1/2 (NC 2 /7io) N )] = (2w/u) N ) 1 /2-hk 2 /7r WN M 
* = Ey/fic, this becomes 

[(2w/w N ) 1/2 A](E r /hwN)( E r / M c 2 )«(l/2)(w/w N )V 2 x 10 4 x 10" 4 

«(l/2)(w/u) N ) 1 /2 



RESONANCE RADIATION OF NUCLEI 185 

if we use, for example, E y ~ 10 5 ev and "Rc^n ~ 10 ev - Thus the 
central peak becomes relatively rather high as the line becomes nar- 
row. The suppression of this peak at higher temperatures is dis- 
cussed in the next section. The line is undisplaced by recoil. This is 
possible (despite the physical expectation that the crystal should re- 
coil as a whole) because in the Fourier analysis (1) it is assumed that 
the ends of the string are fixed. 

This simple remark about the height of the central peak only ap- 
plies, however, if the springs are stiff enough so the zero-phonon 
transition may be evaluated by Eq. (21') for all s values down to 
s = 1, that is, that the dividing line between D x and D 2 becomes n = 1 
and that none of the factors fall into D x . This requires that b\ be 
considerably less than unity. For a string of beads Nw s = soj^, and 
b| may accordingly be written in two ways, b| =1ik 2 /sMwj s j or 
b| = - nk 2 /NMu> s . The requirement b 2 = "nk 2 /Ma>N < 1 * s ver Y nearly 
satisfied by the sample numbers (10 4 x 10" 4 ) given above. If it is not 
satisfied for s = 1 but the corresponding requirement is satisfied for 
s = 2, that is, if b^ 1, or only slightly less, and b 2 is well below 
unity, then D x contains a factor for s = 1 and D 2 contains no-phonon- 
transition factors about equal to unity for all values of s. 

This provides a simple illustration with small quantum numbers, 
similar to the more general discussion above of Eq. (20') for large 
quantum numbers. Even here in this simple case we have the compli- 
cation that D x can have a value not only for the one-phonon jump but 
also for the sever al-phonon jumps of n lf from to several values of 
n x . We may simply evaluate the first few transition probabilities D 2 
by explicit use of the normalized Hermite polynomials. 

Ho = V V* H 2 (|) = 2 1 / 2 7r -V4(|2 _ 1/2) 

H^) = 2 1/2 iT 1 / <l I HgU) = 3" V2 ^-1/4(2^2 -3|) 

From Eq. (20') we have 

IDjI = e -U/2)b? i/h^, [, . (i/2) ibl ] e"" 2 dr?| 2 

and find that the successive transition probabilities for the zero-, 

_(1 /2)b2 
one-, two-, and three-phonon jumps are e 1 multiplied by 

1, (l/2)b 2 , (l/2)(b 2 /2) 2 , and (l/3)(b 2 /2) 3 , respectively. This is again 

a question of obtaining an approximate match between the peri- 

odicity of e s and of H ng ,(£) so as to destroy as effectively as 
possible the orthogonality between the latter and H . For b x = 1, we 
see that the strongest peak of the spectrum is still at the undisplaced 
position; then, half as strong as that, there is the one-phonon peaK at 
the s = 1 position in the sequence of Fig. 2, and then, one- eighth as 



186 RESONANCE RADIATION OF NUCLEI 



strong, there is the two-phonon peak, which happens to be near the 
s = 2 line in Fig. 2 because the lines are almost equally spaced. With 
h\ = 2, the zero-phonon peak is reduced to equality with the one-pho- 
non peak and, as b x increases, the highest intensity moves further out 
through several phonons. As the spring grows weaker and b x increases 
further, the other modes s = 2, 3, etc., start to contribute to the inten- 

_(l/2)b2 
sity of the "wings" in a similar fashion and their factors e s 

for the no-s-phonon jump further detract from the intensity of the 
zero-phonon peak. 

Thus, in this very simple system we see with especial clarity that 
we first start to get phonon excitation to detract from the intensity of 
the central peak when the originally very stiff springs become weak 
enough to satisfy the condition b^» 1. The momentum carried away 
by the y-ray is fik, and this condition (see previous page) may be 
written either as 

(tik) 2 /2NM = (1/2)110)! 
or in the form 

(fik) 2 /2M = (l/2)fico N 

The first form suggests that we have the proper "impedance 
match" when the kinetic energy of the recoil (provided the recoil mo- 
mentum is divided among all the N atoms which move in phase with 
each other in the first vibrational mode) is about equal to the kinetic 
energy involved in the one-quantum excitation of this mode. (The 
specification of kinetic energy is here not essential, for there are 
still factors 2 floating around depending on such details as whether 
or not nucleus R is at the middle of the string, R = (1/2)N, to give it 
maximum "leverage" for this and other odd modes.) This is perhaps 
the most graphic physical explanation of the sharp central peak: when 
the y-ray momentum is less than enough to supply this collective mo- 
tion of the atoms in the middle relative to the ends of the string, there 
is apt to be no phonon excitation. 

In our simple model, with the ends of the string fixed, the central 
peak is undisplaced by recoil. In a more realistic model, we could 
merely constrain the end atoms (by a light frame) to remain a fixed 
distance apart and to conserve total momentum. The central peak 
would then be slightly displaced by recoil of the whole system and the 
s = 1 line would be further displaced by this additional recoil [if 
R^ (1/2)N] arising from excitation of the first one-phonon jump. 

The second form of the condition shows that the recoil momentum 
of the free nucleus, not shared with any others, corresponds to an en- 
ergy about great enough to excite even the highest-frequency mode of 
vibration. The interesting fact is that high-frequency modes are nev- 
ertheless not excited. One might say that there is an impedance mis- 



RESONANCE RADIATION OF NUCLEI 187 



match, or that there is no way for a gradual transfer of momentum to 
the vibratory system via nucleus R to excite the various nuclei in 
their high-frequency motions with contrary phases. If nucleus R 
should shoot off a bullet with this much momentum at a given time, 
classically, for any finite stiffness of spring, the nucleus and bullet 
act as an isolated system with no time for transfer of momentum to 
them during the instant of ejection. The full free recoil of the nucleus 
would be detracted from the energy of the bullet, and the energy of re- 
coil would first excite a localized traveling wave packet which could 
then be expanded in terms of various fairly high frequency vibrational 
modes with specified phases. If the y ray we have discussed were 
emitted at a given instant, this energy would be sufficient to excite a 
localized wave of much higher amplitude than the one -quantum ampli- 
tude of o>n- Corresponding to the expansion of the localized wave, 
there should be perhaps a one- or several-quantum excitation of a 
few of the moderately high frequency waves. Such excitation does not 
occur with y-ray emission because the emission does not take place 
at a given time. Instead, it takes place at an unknown time with a prob- 
ability which builds up very gradually in a coherent manner as we have 
seen in Eqs. (19) and (20), and, in the case discussed, it may either 
excite a low-frequency phonon or none at all. 

Discussion of the Approximation in Expanding the Exponentials 

The insufficiency of the approximation in expanding exponentials 
is apparent in the possibility that the diagonal element, Eq. (210, may 
go negative even with all the n s = 0, and a fortiori at a higher tem- 
perature. In one dimension, if we use the approximation and constants of 
Eq. (160, we have 

N 
(1/2)7] b| « (3/4) /(ds/s) = (3/4) In N > 1 (26) 

s 

Here we have used a factor 1/2 from a mean value of sin 2 (Rs/N) of 
Eq. (16). In three dimensions the corresponding sum is 

N 
(1/2) £ b| tu « (3/4N 2 )(tt/2) / (p 2 dp/p) = Sir/16 
stu 

p 2 = g 2 + t 2 + u 2 ( 26 /) 

which is still of order unity, although the importance of the long 
waves has been suppressed. We should therefore examine the expan- 
sion of the exponentials more carefully. 

In place of the square bracket representing the high-frequency 
modes toward the end of Eq. (16), we may write a product of expo- 



188 RESONANCE RADIATION OF NUCLEI 

nentials and for its matrix element in place of Eq. (21) we may write 

|D 2 | 2 ~<n n -4j|e lbs ^ s |n n ...n N >| 2 =| n <n^| e lbs ^ | n s >| 2 

s^n 

* |n<n' s |1 + ib s | s - (l/2)b 2 s || - |n s >| 2 

= |n[l+ib s <n' s |£ s |n s >- (l/2)b|<n^|| 2 | n s >-]| 2 (27) 

Considering now only the diagonal element, ng = n s , we have matrix 
elements of only the even powers of £, as follows: 

|n s [l-(l/2)b 2 s <n s |4||n s >+ (l/4!)<n s ||||n s >+ -]| 2 (28) 

This product is of the form 

n s (l + e s ) = 1 + ^s + £s*t e s£t/ 2! + ^r* s* t e r E s et/3! '" 
which differs from the exponential 

e £e s = 1 + Lz s + (Le s ) 2 /2\ + (Ee s ) 3 /3! ••• 

only by about N of the N 2 terms in e| and by about N 2 of the terms 
in s|, etc., leaving us one order better off than one might otherwise 
expect. We can now in this approximation put 



|D 2 | = e -Z]b 2 s<ns|l 2 s|n s > + (2/4!)Z:b 4 s <ns|l||ns>+ •■ 
„ -£b 2 s <n s |4 2 s |n s >_ -£(n s + l/2)b| 



(29) 



This replaces Eq. (21') and is clearly more satisfactory in that it in- 
dicates a gradual reduction of the intensity of the central peak to zero 
as the sum increases. 



Shape of the Wings in One and Three Dimensions 

The derivation is slightly modified for two or three dimensions. 
We are of course interested in three, but shall write the equations in 
two dimensions for the sake of economy of symbols. Each component 
of the vector displacement of atom qr can be described in normal co- 
ordinates, but we shall write only the x component of such a displace- 
ment. (The Lagrangian also contains the other components independ- 
ently.) The force constant between adjacent atoms, owing to a relative 
displacement in the x direction, is a longitudinal force constant, K, 
if they are neighbors in the x direction, a transverse one, K transv 



RESONANCE RADIATION OF NUCLEI 189 

= fK, if they are neighbors in the y direction. The Lagrangian is 

£ = (M/2) L*? qr - (K/2a) L [(x q+ 1§ r - x q>r ) 2 + i(x q>r + 1 - x q>r ) 2 

= (M/2) £q| t - (K/a)Z!q 2 t {[l - cos(tts/N)] +f[l - cosfrt/N)]} 

(20 
where 

x qr = (2/N) 1 ^ 2 ]T q s t sin (qsir/N) sin (rtir/N) v = 2 

st 

Taken for all lattice points qr, this is a wave of x displacement hav- 
ing q nodes in the x dimension and r nodes in the y dimension. The 
frequency of a normal mode is given by 

w| t = (2K/Ma){[l - cos (tts/N)] + f[l - cos (?rt/N)]} 
« (K/Ma)(7r/N) 2 (s 2 + ft 2 ) 
The perturbation term 3C' contains a factor [see Eq. (16)] 
e ikx QR = n st e ib st*st 
b st = (2/N)*^ 2 (VMw st )V 2 k sin(qs7r/N) sin (rt7r/N) (16") 



or, averaged over the factors sin 2 « (1/2), in the neighborhood of a 
given st, 

Wt = N - " (ii/Mw st ) k 2 » (3/2N^ " V(s 2 + ft 2 ) 1/2 

or, in three dimensions, 

b| tu = (3/2N 2 )/[s 2 + f (t 2 + u 2 )] 1/2 (16"') 

In three dimensions, the number of modes with cu s t between wand 
it) + doois N(co) dw, with 

N(w) = (N 3 /27r 2 )(Ma/K) 3/2 co 2 (30) 

for the simple extreme case f = 1. For the other extreme, f = (no 
shear strength), we have instead 

N(w) = (NA)(Ma/K) 1/2 (220 

independent of u>, just as in the one-dimensional case. For f = I, the 



190 



RESONANCE RADIATION OF NUCLEI 



shape of the wings (in the lower-frequency part of the spectrum) is 
given by 



[n s + (l/2)]b 2 s N(w s ) 



to). 



n s = ° 



const., n s + l/2«n s = kT/nw s (24') 



as indicated by the broken line and the solid line, respectively, in 
Fig. 4. 




Fig. 4. Spectrum for a three-dimensional lattice. The outer peak 

corresponds to the upper limit of the (oversimplified Debye) 
spectrum for compression waves, the peak at fcojsj similarly 
corresponds to the upper limit for shear waves. 



Here we see the real source of the difference between the behavior 
in one and in three dimensions. In one dimension, there is a low- 
frequency divergence in the shape of the absorption wing. The same is 
true in three dimensions without a consideration of shear strength. In 
the latter case one has justification for the intuitive feeling that the 
problem is mainly one-dimensional in its essence, because the photon 
excites vibratory motion along its direction of incidence. However, in 
three dimensions with shear strength taken into account, the frequency 
of this longitudinal vibration of the absorbing atom is affected by the 
transverse variation of the wave amplitude. There are so many possi- 
ble wave numbers in the two transverse dimensions that there are 
many more high-frequency than low-frequency modes. Thus, the low 
frequencies are effectively suppressed, and the shape of the wing, 



RESONANCE RADIATION OF NUCLEI 191 

rather than going up as Wg 1 as the central spike is approached, is 
linear in w s at zero temperature. For finite temperature, it is con- 
stant over the low -frequency part of the spectrum for three dimen- 
sions with shear strength, as compared with the w s 2 dependence for 
one dimension. 

Meaning of the Sharp Peak 

Perhaps the greatest surprise that one encounters when thinking 
intuitively about the dynamics of the crystal motion is that there can 
exist such a very sharply defined component of the absorption as the 
central spike, so nearly exactly at its undisplaced position, in spite 
of the complexity of the motion of the absorbing nucleus and in spite 
of the normal process of recoil, which for a free atom gives a very 
appreciable Doppler shift. Between free nuclei the recoil Doppler 
shift is quite enough to throw them out of resonance, unless they have 
an initial velocity toward each other to compensate, whereas between 
the nuclei in solids of Mossbauer's experiments the resonance is 
quite exact when the solid bodies are at rest, and is destroyed by a 
relative velocity of a few centimeters per second in either direction. 

In this analysis we see how it is possible, in principle, for the cen- 
tral line to be quite sharp and almost undisplaced, even in the pres- 
ence of a considerable amount of excitation of the lattice vibrations. 
The sharpness is a typical quantum effect, like the sharpness of 
atomic spectral lines that was not expected classically. It arises 
from the fact that the crystal vibrations are quantized and that they 
may not receive an arbitrarily small amount of excitation energy ex- 
cept in arbitrarily large wavelength. There is indeed a small broad- 
ening and shift due to the long waves. If the process of gamma ab- 
sorption or emission were to be thought of as a completely sudden 
process, giving the nucleus a sudden kick in a time much shorter than 
a lattice vibration, one would expect this impulsive initial motion to 
show up as a recoil Doppler shift, as it does in a free atom, and sub- 
sequently to have its energy divided between the various lattice modes 
in a manner equivalent to a Fourier analysis of the impulsive velocity. 
Actually, as we see in the theory, the possibility of transition is not 
determined merely by the energy and instantaneous momentum bal- 
ance, but rather the probability of transition is built up by integration 
over the mean life of the nuclear state involved. This is time enough 
for ample transfer of momentum between the nucleus and the lattice 
by way of the forces involved in the lattice vibrations. In some cases, 
the recoil momentum is divided equally between all the atoms of the 
crystal, with no phonon excitation, and the corresponding contribution 
to the sharp line is practically undisplaced by recoil. With phonons 
initially excited and the radiating nucleus thus in a vibrational motion, 
the transition involving no additional phonon excitation continues to 



192 RESONANCE RADIATION OF NUCLEI 

occur. For this transition, the to-and-fro motion of the radiating nu- 
cleus does not make any Doppler contribution to the character of the 
radiation, even though it would classically. f This is exactly analogous 
to the fact that the to-and-fro motion on the charge on an electron in 
a Bohr orbit does not contribute to the radiation without a quantum 
jump, even though it would contribute classically. For high phonon 
quantum numbers, the one-phonon transition between states of the vi- 
brational motion should correspond to a classical result, just as the 
An = 1 quantum jump does in the more familiar correspondence- 
principle discussion of the radiation from large Bohr orbits. As the 
temperature is raised well above zero, the wings we have described 
thus have a close correspondence to the classical "side bands" from 
a moving source. As in the classical case, the wings merely detract 



f The classical Doppler contribution due to a sinusoidal motion 
x = a sin co m t of a radiating source is in the form of weak "side 
bands." These one obtains analytically from the Doppler -shifted fre- 
quency co(t) = co [l + (act) m /c) cos co m t] in the expression exp (i / o> dt) 
by simply treating act> m /c as small in expanding the exponential and 
by expressing sin u> m t in terms of exponentials. In physical terms, 
the appearance of a strong central line o> flanked by weak lines at 
u) Q ± co m may be understood by thinking of the effect of this slightly 
varying co(t) on a tuned detector. At the end of a swing, co(t) becomes 
instantaneously stationary at co ± oj m and comes back in phase at this 
frequency at the end of the following swing, so that a detector will re- 
spond to these frequencies. Now consider a detector tuned to oj . If it 
is in phase with the signal just when co(t) = cu and if co(t)> co for the 
next half- cycle, then the signal will gradually get ahead of the detec- 
tor in phase during this half- cycle. If the cycle is short enough so that 
the phase lead remains less than tt/4, there will be no destructive in- 
terference (and, if less than tt/2, there will still be a preponderance 
of constructive interference). The unique feature of w is that for it 
alone this advance in phase will be exactly compensated for by the re- 
tardation in the following half-cycle, so that the response will con- 
tinue to build up, being fed by most of each later cycle and giving a 
strong line. The condition that the phase lead remain less than 7r/4 is 
a<7rc/4w = A/2. The y-ray wavelength A = (h/mc 2 )(mc 2 /137E J is 
much smaller than the atomic unit of distance h/mc 2 , and the inten- 
sity of the central line dies away as the vibrations of the atoms in the 
crystal attain an amplitude larger than A. The corresponding condi- 
tion for zigzag motion over a distance a is a < A/4. This may be un- 
derstood from the picture used in the usual graphic explanation of the 
Doppler shift, showing an original wave train and one compressed into 
a shorter space by the motion of the source. The detector tuned to re- 
ceive the uncompressed wave receives the compressed wave instead. 



RESONANCE RADIATION OF NUCLEI 193 

from the intensity of the central line and do not widen it except, even- 
tually, by making it disappear in the wider background as the proba- 
bilities of phonon transitions increase with the pre -excitation of many 
modes. 

Acknowledgments 

I am indebted to Dr. Maria G. Mayer and Dr. H. J. Lipkin, partic- 
ularly for discussion of the situation with three dimensions and of the 
improved expansion of the exponentials, also to Dr. John Schiffer for 
informing me of the experimental phenomenon. 



194 



Resonance Absorption of Nuclear Gamma Rays and the Dynamics 
of Atomic Motions* 

K. S. SlNGWI AND A. SjOLANDER 

Argonne National Laboratory, Argonne, Illinois 
(Received June 13, 1960) 



The theory of resonance absorption of nuclear y rays is generalized for an arbitrary system of interacting 
particles by expressing the relevant transition probability in terms of a space-time self-correlation function ; 
and thus relating the resonance line shape to the incoherent differential scattering cross section for slow neu- 
trons. Two limiting cases : (i) a gas and (ii) a solid have been considered. Discussion regarding the justifica- 
tion of the use of a classical self -diffusion function for a liquid is given and expressions for the broadening of 
the resonance line due to diffusive motions of the atoms of the interacting system are derived. It is suggested 
how Mdssbauer-type experiment could be used to give information regarding the diffusive motions of atoms 
in a solid and also, under more favorable circumstances, in a liquid. 



INTRODUCTION 

THE observation by Mossbauer 1 that nuclear y 
rays can be resonantly absorbed or scattered by 
nuclei bound in a crystal lattice has recently led to some 
very interesting applications 2 and holds promise for 
more applications particularly in the field of solid-state 
physics. Mossbauer's observation rests on the fact that 
in the case of a nucleus bound in a crystal, a y ray can 
be emitted or absorbed without any energy transfer to 
and from the lattice. The probability of such a recoilless 
transition is, in most cases, small and is governed by the 
usual Debye- Waller factor, familiar in the theory of 
x ray and neutron scattering. Mossbauer explained his 
experimental results on the basis of a theory due to 
Lamb 3 for the Doppler broadening of neutron absorp- 
tion resonance. Both in the theory of neutron and y- 
ray resonance absorption the relevant matrix element 
corresponding to a transition of the crystal lattice from 
one state to the other is the same. 

The purpose of this paper is two-fold : one is to gen- 
eralize the theory for an arbitrary system of interacting 
particles by expressing the transition probability in 
terms of a space-time self-correlation function, which as 
is well known, determines the incoherent scattering for 
slow neutrons; and the second is to show how Moss- 
bauer technique can be used to gain information con- 
cerning the nature of diffusive motions of atoms in a 
solid and also, under more favorable circumstances, in 
a liquid. The cross section for 7-ray resonance absorp- 
tion in the case of a gas (Bethe Placzek formula in the 
case of neutrons) and in the case of a solid, in the limit 
of both weak and strong binding (two limiting cases 
of Lamb's theory in the case of neutrons), follows very 
simply from one general formula. Furthermore, the 



* Based on work performed under the auspices of the U. S. 
Atomic Energy Commission. 

■R. L. Mossbauer, Z. Physik 151, 124 (1958): Naturwissen- 
schaften 45, 538 (1958); Z. Naturforsch 14a, 211 (1959). 

2 During the last year and this year a number of communica- 
tions concerning the Mossbauer effect and its various applications 
have appeared in the Physical Review Letters to which the reader 
is referred. 

» W. E. Lamb, Phys. Rev. 55, 190 (1939). 



generalized formula can be of great help in more 
complicated systems as for instance liquids, where it is 
difficult to treat the dynamics of atomic motions in 
detail. 

MATHEMATICAL FORMULATION 

We are interested in calculating the probability of 
absorption or emission of a 7 ray of momentum p by a 
single nucleus of an interacting system (say solid or 
liquid) such that the nucleus makes a transition from a 
state A to a state B and at the same time the interacting 
system makes a transition from a state, say | nO) to a 
state I n). Since the interaction within a nucleus is much 
stronger than that between two nuclei, the total wave 
function can be written as a product of wave functions 
one of which depends only on the coordinates of the 
centers of masses of different nuclei and the other de- 
pends on the coordinates of the nucleons relative to the 
centers of masses of their respective nuclei. The transi- 
tion matrix element, corresponding to the absorption of 
a photon, can be written as (Bn\H'\nOAp), where H' 
represents the interaction between the radiation field 
and the nucleus and has the following form : 

H'=Y,ica 9 exp (ip • ti/h) 

= exp(z'p- Ra/h) H« ca p exppp- (r,— R a )/A]. 

c is a constant depending on p, a v is the annihilation 
operator for a photon with momentum p, r, is the co- 
ordinate of a nucleon of the nucleus a, and R a is the 
coordinate of the center of mass of the nucleus. The 
interaction operator H' is thus a product of two terms, 
one of which depends only on the coordinates of the 
nucleons relative to their center of mass and the other 
depends only on the coordinates of the center of mass. 
Thus the matrix element of the transition is a product 
of two matrix elements, one of which corresponds to the 
change in the internal state of the nucleus and the 
other is («|exp(ip-R o /Ai)|«0), corresponding to a 
change in the state of the collective motions of the 
centers of masses. The first matrix element is just a 
constant for our purpose, and it is the second one with 
which we shall be mainly concerned here. It then follows 



1093 



195 



1094 



K. S. SINGWI AND A. SJOLANDER 



from the usual dispersion theory 4 that the absorption and |»0) of the interacting system, T is the natural 

cross section per nucleus for a y ray of energy E is width of the excited state of the nucleus and g n0 is the 

given by statistical weight factor for the state |«0). In Eq. (1) 

<r r 2 I (n I exp (ip • R/h) \n0)\ 2 tne nuclear width T has been assumed to be independent 

a a {E) = E gno , (1) 0I tne state | n). Also the suffix a in R has been omitted. 

4 n.Tto (E — E+e n — e n o) 2 +r 2 /4 The constant before the summation sign has been so 

chosen that a a (E) goes over to the familiar Breit- 

where Eo is the energy difference between the final and Wigner formula for a fixed nucleus, a being the reso- 

the initial nuclear states of the absorbing nucleus, t„ nance absorption cross section. 

and e„o are, respectively, the energies of the states \n) Now Eq. (1) can be written as follows: 



<Ta(E)- 



r r 2 



4 

cr r 2 i 

4 



E g„o|<n|exp(ip-R/ft) 



«0)| 2 f 



«[>-(e»-e»o)/ft] 



dp 



(E-E -h P y+T*/4: 
— f dt{Z ^o|<n|exp(i P -RA)KO>| 2 expP/(6 n -e n o)/^]} 
exp(-i*p) 



-dp 



J_. (E-E -hp) 2 +T*/4 
= — I txpl~U(E-Eo)/h- (T/2h) \t\Jdt 
X[ E gno(nO\ exp(-ip- H/h+itH/h) \ n)(n\exp(ip-R/h-itH/h) \ «0>] 

n,n0 

<r r /•* 
= — I exp[-t7(£-£o)/*-r/2*| /| ]<exp[-*p- R(0)/»] exppp- R(/)/*]>t*, 
4A •/ _ 



(2) 



where R(*) is the Heisenberg operator denned by 

R(0 = exp(afl/ft)R exp(-UH/h). 

H being the Hamiltonian of the interacting system, and 
<• • -) T means both the quantum mechanical and the 
statistical average at temperature T. We shall here 
restrict ourselves to a system for which Boltzmann 
statistics is applicable. In the above derivation the 
Fourier representation of the 8 function and the identity 
En|»)(»| =1 have been used. We now define a func- 
tion G,(x,t) through the following equation 

(expd-ip- R(0)/ft] exppp- R(fl/*]> r 



/ 



exp(tp-r/ft)G,(r,/)<Zr. (3) 



The inversion then gives 

G,{tfy- (It)- 3 f exp(-tp-r/ft) 

X(exp[-ip- R(0)/*] exppp- R{t)/K]) T d(?/h) 

= ^Jjr'5[r+R(0)-r']5[r'-R(/)]) . (4) 

4 W. Heitler, The Quantum Theory of Radiation (Oxford Uni- 
versity Press, New York, 1944), 2nd ed., p. 110. 



For t=Q, G,(r,0) = 5(r). G,(t,t) describes the correlation 
between the position of one and the same particle at 
different times. It gives, in the classical case, the proba- 
bility of finding a particle at time / at position r, if the 
same particle was at the origin at time t=0. The 
interpretation of this function is not quite clear in a 
quantum mechanical treatment and is discussed in the 
Appendix. Van Hove 6 has discussed the G,(r,t) func- 
tion in detail and we shall refer to his original paper. 
From (2) and (3) we have 



,(£) = 



cor r 

— I expp(K-r-c 

4h J 



*)-(T/2ft)|*|] 

XG,(i,t)dTdt, (5) 
where hw=E—Eo, #tc= p. 

As shown by Van Hove, 5 the incoherent differential 
scattering cross section for slow neutrons is propor- 
tional to the integral in (5) with T=0. In Lamb's 
theory, 3 the probability of resonance absorption of 
neutrons of energy E is also given by Eq. (1) besides a 
constant factor and is, therefore, proportional to the 
integral in (5). Thus the relevant term in the cross sec- 
tion for all the three processes — resonance absorption 
of neutron and y rays by nuclei and neutron scattering 
(with T = 0) is given by an expression of the type (5). 
We shall see in the sequel that the cross section for the 



'L. Van Hove, Phys. Rev. 95, 249 (1954). See also R. J. 
Glauber, Phys. Rev. 98, 1092 (1955). 



196 



RESONANCE ABSORPTION OF NUCLEAR GAMMA RAYS 



1095 



resonance absorption of y rays by nuclei whether in a 
gas or bound in a solid or a liquid will follow from (5) 
depending on the explicit form of G e (r,t). It is possible 
to calculate the function G,(T,i) rigorously in the case 
of a Maxwellian gas and in the case of a solid in the 
harmonic approximation but it is not possible to do so 
in the case of a liquid. Nevertheless, in the latter case 
one could use in an approximate way the classical form 
of G,(r,t) ; e.g., the solution of the usual diffusion equa- 
tion or better the solution of Langevin's equation for 
Brownian motion. The behavior of G,(r,t) for very 
small and very large times is known and for inter- 
mediate values of the time one could try different forms 
of G,(r,t) so as to fit the experimental data. Thus, a 
general formulation of the absorption probability (the 
same holds for emission) in terms of the self-correlation 
function G,{x,t) as expressed by Eq. (5), has a definite 
advantage. 

It has been shown by Van Hove 6 that the self- 
correlation function in the case of a gas or a solid (cubic 
symmetric crystals) has the general form 



G a {r,t) = [2m{t)-]- % exp[-rV2 T (/)]. 



(6) 



There is no obvious reason to believe that in the case 
of a liquid G a (r,t) has also the above general form. It 
seems, however, reasonable to assume that (6) is a 
good approximation for a liquid too. We know that it is 
correct for small as well as large times. 

The probability w e {E) for the emission of a y ray is 
also given by Eq. (1) except that the signs of e„ and e„o 
are interchanged and the constant is different. 4 Pro- 
ceeding as before, it is easy to show that w e {E) is given 
by 

w t (E)= f exp[i(K-r-wt)-(T/2h)\tn 

2irhJ 

XG,(t, -t)dtdt. (7) 

It is normalized such that Jl x w e {E)dE=\. 

The quantity of experimental interest is the self- 
absorption cross section <x which for a thin absorber is 
defined by 

o= f <r a (E)w e (E)dE I f w e (E)dE 

= f a a (E)w e (E)dE. (8) 

In a y-ray resonance absorption experiment if the 
emitter is made to move with velocity v relative to the 
absorber, the emitted y ray gets an energy Doppler 
shift s=(v/c)Eo, c being the velocity of light, and in 
that case the argument of a a {E) should be replaced by 
s-\-E. If we do this and make use of (5), (6), and (7) in 
(8), it follows that the self-absorption cross section is 
given by, noticing that the integration over E can be 



extended to — °o without any appreciable error, 
(7 r r / its T 



is) 



=— I exp( 1*|) 

4h J \ h k / 



h h 
Xexp 



-Ct.(0+t.(0]*- (9) 

2 J 



And if the emitter and the absorber are identical (9) 
becomes 

«r.(*)=W), (10) 

where <r a (s) is given by (5) with T replaced by 2T and 
y(t) replaced by 2y(l). Before we proceed to calculate 
<t(s) for a nucleus bound in a solid, we shall evaluate the 
absorption cross section a a {E) for a nucleus in a perfect 
gas. 

ABSORPTION OF A y RAY BY AN ATOM 
IN A PERFECT GAS 

It has been shown by Van Hove 6 that for a perfect 
gas the quantum mechanical form of G,{r,t) is the one 
given by (6) with 

ht k B TP 

•^-V-TT (11) 

where M is the mass of the atom and ks is the Boltz- 
mann's constant, and T is the temperature of the gas. 
The term linear in t is purely of quantum mechanical 
origin and the term quadratic in t survives in the classi- 
cal limit. Vineyard 6 has shown that for very small times 
for an arbitrary interacting system described by a time- 
independent Hamiltonian y{t) is given by 



ht 1 
M 3M 2 



(12) 



where p is the momentum operator of an atom. This 
result easily follows from (3) if we expand the operator 
R(0 in powers of t and define y(t) by the relation 

y(t)=-fr*G.(r,t)dT, 

and assume that the system is isotropic. 

Substituting (11) in (6) and using the resulting G 8 (r,t) 
in (5) and after performing integration over r we have 



Va{E) 

<r r 

where we have put 

R (the recoil energy) = h W/2M=E 2 /2Mc\ 

A = 2(Rk B T)*. 

• G. H. Vineyard, Phys. Rev. 110, 999 (1958). 



' /•« r it r A¥i 

- exp — (E-Eo-R) 1/| k 

J_ L h 2h 4#J 



(13) 



(14) 



197 

1096 K. S. SINGWI AND A. SJOLANDER 

Putting y=tT/2h and making use of the convolution atomic cubic crystal y(t) is given by 
formula for the Fourier transform of a product, we have >o 

* r w r i z \( zt \ "T/W 

!=— I cothl J I 1 — cos- I— i sin- <fe, 

34V L \2k B T/\ k/ k] z 



*.(£}<— I & 

.. » r (F—E —R) A 2 n l where f(z) is the distribution of energy levels of the 

X I f exp -iyz-iy r 2 — \dy phonons and such that 

1(2t)»J_ L r/2 Pj I 

fir" ] I f(z)dz=l. 

which gives /( z ) is zer0 beyond 2=2^. 

,. ot, , Equation (5) with the help of (6) can be written after 



f /«- az . -1—-- ,-,-■-- 

<r a (£)=<r I exp[-^(8+x) 8 ], (15) integration over r as 

2v^ 1+z* 
where °^ ' 

3p[-i**y(«)]J exp(-*orf-— 1/|)<« 



E-E -R T <T rt 

x= ; |=-. (16) U 

T/2 A 4AI 



"'r.; +e*p[-i«M»)]£e*p(-^|,|) 



Formula (15) is the same as given by Bethe and 
in the case of resonance capture of neutrons 
forming a Maxwellian gas. 

X{exp[J««(7(co)-7(0)]-l}*l, 
RESONANCE ABSORPTION OF y RAYS BY ' 

ATOMS IN A CRYSTAL , . . . , , , , . m 

where 7(00) is the value of y(t) at t= ». The exponen- 
We shall here restrict ourselves to cubic Bravais tial within the square brackets can now be expanded 
lattices. Again Van Hove 6 has shown that for a mono- in a power series, and we have 

<t t rftr 1 - (««/2)» 1 r » / r \ 

(T a {E) = 2ir — exp(-2W0 +I> j expl -tut l/l jDyC 00 )-^)]"* 

4» l27r(£-£o) 2 +rV4 «=i »! 2»J_ V 2A ' / 

WTr 1 . (2in» i 

= +L gn(E-E , T) exp(-2W0, (18) 

2 L2tt (£-£o)»+P/4 tx n \ * J 



since T can be neglected in the integrand in Eq. (18), tive to compare formula (18) with that for the inco- 
and where herent differential scattering cross section for neutrons 

s (E T)= i cothl I — ll /in\ m solids. The first term inside the curly brackets gives 

2EF{T)Y \2/&b7V J' a sharp absorption peak of width T and represents the 

recoilless absorption of 7 rays, and corresponds to 



(ET)= f e (E—E' Tie (E f T)dE' C20") elastic scattering in the neutron case (where it is a delta 

/_, > n- , , function). The other terms give a broad peak extending 

at least over an energy z max and corresponds to phonon 

F(T)= f f — coth(—)dz, (21) exchange. _ _ 

J z \2&fl7/ In the Debye approximation, 
and 

ftV /(z) = 3z 2 /(/b0) 3 for z<£ B ,„. 

2^=|^(=o)=— F(T). (22) =0 for z>/feB e, (23) 

2W is the usual Debye-Waller factor. The above formu- @ bein S the Deb y e temperature of the solid. Using 

lation is the same as that used by Sjolander 8 in con- ( 23 ) in ( 21 ) ll 1S eas y to show that 

nection with neutron scattering by solids. It is instruc- 

6 rl T /0\-j 

J H. Bethe and G. Placzek, Phys. Rev. 51, 462 (1937). F{T)= H 01 — J , (24a) 

8 A. Sjolander, Arkiv Fysik 14, 315 (1958). Jfe B 0L4 V T/ J 



198 



RESONANCE ABSORPTION OF NUCLEAR GAMMA RAYS 



1097 



where 



1 r* ydy 

♦to— I — ■ 

zJ e v -l 



The function <f>(z) has been tabulated by Zener. 9 Now 



F(T) = 3/(2k B ®) for T«0 
= 6T/(k B @ 2 ) for 7»0. 

We also notice that 



(24b) 



/; 



UE)dE=: 



Experimentally we are interested in the self-absorption 
cross section; i.e., in a(s) given by (9). Here the emitter 
and the absorber are assumed to be of the same ma- 
terial but at different temperatures, say T e and T a , 
respectively. Let v be the relative velocity of the emitter 
and absorber. s= (v/c)Eo is positive f the two move 
towards each other and negative if they move away 
from each other. Proceeding as before, it follows from 
(9) that 



Us)\ 



t<tqT 

2 

rr 1 - (2W 4 +2W a ) n 

X +E Sn(s)\, (25) 

UyH-r 2 —l «! 

where 

2W e = (hW/2M)F(Te), 
2W a = (hW/2Af)F(T a ), 

1 

gi(s)= 

F(T e )+F(T a ) 

XZF(T.)gi(s,T.)+F(T a ) gl (s,T a )], (26) 



9n(s)= I Siis—^g^-iWdY. 



It should be borne in mind that the energy distribution 
of the phonons changes with temperature and hence the 
Debye temperature also changes. There will be a very 
slight shift of the resonance absorption peak due to the 
fact that when a y ray is emitted or absorbed the mass 
of the emitting or the absorbing nucleus changes. This 
second order Doppler effect 10 has not been considered 
here. 

If the emitter and the absorber are both identical 
and are at the same temperature, Eq. (25) simplifies to 



2 



l7T5 2 +r 2 



(4W0" 1 

— g„(*,DJ (27) 



+r 2 n=i n! 



The first term in (27) gives a sharp resonance peak 
having a full width 2I\ For s=0, the first term is 



er'(0) = §o-o<r 



(28) 



The resonance self-absorption cross section is thus 
diminished by a factor e~ iW , where 2W, in the Debye 
approximation, is given by Eqs. (22) and (24). From 
the expression for 2W, it is clear that to have a large 
Mossbauer effect the recoil energy R of the nucleus 
must be small; i.e., the y ray should have a low energy 
and the Debye temperature of the solid should be 
large and the temperature as low as possible. Recently 
Lipkin 11 has derived the expression for 2W in a simple 
manner and has also come to the same conclusions. In 
the original Mossbauer experiment in which Ir 191 129- 
kev gamma rays were used (R = 0.046 ev, &s0 = 0.025 
ev, r=88°K), 2W was nearly equal to 3; and hence the 
resonance effect was very small. For a large resonance 
effect <r'(0) has to be greater than the non-nuclear 
cross section such as the cross section for the photo- 
electric effect. Unfortunately for isotopes so far in- 
vestigated, the Mossbauer effect is very small except in 
the case of Fe 67 , where because of the low 7-ray energy 
(£o=14.4 kev), 2W is nearly 0.1 at T=0*K, and this 
is the reason why it is possible to observe the Moss- 
bauer effect even at very high temperatures. Because 
of this comparatively large effect and the extreme sharp- 
ness of the resonance line (r~5XlO- 9 ev), the Moss- 
bauer effect in Fe 57 nucleus has found recently so many 
interesting applications. 

The second term in (27), corresponding to phonon 
exchange, gives the shape of the wings of the sharp 
resonance absorption line ; the wings extending at least 
up to an energy of the order k B Q. If 2W / «1, all terms 
except the first in the sum are negligible. In that case 
the shape of the wings is related in a very simple way 
through Eq. (19) to the energy spectrum f(E) of the 
crystal vibrations. It is thus at least in principle possible 
to measure the energy distribution function of the 
phonons as has been pointed out earlier by Visscher. 12 
In this connection it is important to realize that if the 
nucleus emitting the y rays constitutes a foreign atom 
in a host lattice, what one measures is not the vibra- 
tional spectrum of the host lattice but a spectrum 
which is characteristic of the local surroundings of the 
emitting nucleus. In addition to this, the one-phonon 
cross section is very small and this limits the possibility 
of using the Mossbauer effect to investigate the vibra- 
tional spectra of solids. A better way to study the real 
vibrational spectra is through the use of slow neutron 
scattering. 

If 2H / »1, then we have what is called the weak- 
binding case. For example in the original Mossbauer 
experiment 2W~3 and it falls under this category. 



C. Zener, Phys. Rev. 49, 122 (1936). 
°B. D. Josephson, Phys. Rev. Letters 4, 341 (1960). 



H. Lipkin, Ann. Phys. 9, 332 (1960V 

W. M. Visscher, Ann. Phys. 9, 194 (1960). 



199 



1098 



K. S. SINGWI AND A. SJOLANDER 



From (17) it follows that 

«r.W-hr.<0] 

= -i2R-+2A 2 —+' 
h W 



Doppler width A Eq. (33) for ^ = simplifies to 



for t« (29a) 

k B % 



■R\_F(T e )+F(T a )~\ for /» , (29b) 

k B S 



where 



A 2 = (8/3)22 (£ kin «+£kin a ). (30) 

£ kin is the mean kinetic energy per atom and is given by 
3 



£kin=-f scothf ) 

4J Q \2k B T/ 



f(z)dz 



Forr>0,£ kin «|* B r. 
Now Eq. (9) can be written as 



"(*)' 



' 4h 



\i«v 



Xexpl -i- — \l\--[ye(t)+y a m\, (31) 
h h 2 



where T~fi/k B <d. 

In (31) the exponent ^ 2 [7«(0+7a(0] in the first 
integral can be replaced by (29a), and in the second and 
third integrals by (29b), without introducing appreci- 
able error. It is then easy to show that 



*(*) 



aoTl r 00 f it T 2A 2 1 

= I exp (s-2R) l/l fildt 

tt L 1 J h 4ft 2 J 



2ftr l 

+ exp(-2W e -2W a )\. (32) 

5 2 +r 2 I 

The integral in (32) is the same as the one which occurs 
in (13) and can be evaluated as before. Hence 



*(*)■ 



<r r 2 exp- (2W e +2W a ) 

2 i 2 +r 2 

+(ro _Lr _l_ eX p |: _i r! ( a;+2 )2 ] (33) 

W* J -« 1+2 2 



where we have defined 

£=2r/A, A 2 = (S/3)R(E^+Ev m a ) = Ae 2 +A a 2 , 



(34) 



*= (s- 2R)/T, R= E<?/2Mc 2 . 
If K<1; i-e., the linewidth T is much smaller than the 



«r(0)=— exp(-2W.-2W.) 
2 

ao\/ir r 



2 (A. 2 +A, 



r ^ i 

— exp 

2 )» L A« 2 +A 2 J 



(35) 



Equation (35) is the same as the formula (19) of 
Mossbauer. 13 

DIFFUSION BROADENING OF THE 
RESONANCE LINE 

In this section we shall show how Mossbauer tech- 
nique could be used to investigate the nature of dif- 
fusive motions in solids and probably under more 
favorable circumstances in a liquid too. In a solid the 
slow jumping movement of an atom from one lattice 
site to another gives rise to a broadening of the reso- 
nance line. At ordinary temperatures the broadening 
due to such a diffusive motion is small compared to the 
natural linewidth but at elevated temperatures the 
former can become of the same order of magnitude as 
the latter and even greater. In liquids the diffusive 
broadening is, however, many orders of magnitude 
greater than the linewidth but, unfortunately, the 
resonance absorption cross section is in most cases small 
compared to non-nuclear absorption cross section. This 
makes it difficult to distinguish the resonance line from 
the background. 

The self -correlation function G s (T,t) as defined by 
Eq. (3) is a complex quantity and cannot, therefore, 
easily be interpreted as a self-diffusion function except 
in the case that its imaginary part is negligible. The 
imaginary part, as we know, is quantum mechanical in 
origin. It is, however, possible as Schofield 14 has done to 
define a real function which in the classical case goes 
over to the classical self-diffusion function and which is 
related to the absorption cross section in a similar way 
as is Van Hove's G„(r,t) function. The transformation 
suggested by Schofield and the question of using the 
real part of the Van Hove's G,(r,t) function are dis- 
cussed in detail in the Appendix of this paper. Such a 
discussion besides being relevant to the context of this 
paper is of importance in connection with neutron 
scattering by liquids. 

Following the suggestion of Schofield, if we replace / 
by t+(ih/2k B T),G,(T,t) goes over to F s (t,t), which is 
given by (8A) of the Appendix and y(t) goes over to 
p(t), where pit) is given by Eqs. (9A) and (10A) of the 
Appendix in the case of a gas and a solid, respectively. 
In the Appendix we have derived the expression for the 
resonance absorption cross section a a iE) and for the 
emission probability w e iE), which are, respectively, 
given by Eqs. (HA) and (12A). We shall rewrite them 



13 R. L. Mossbauer, Z. Physik 151, 124 (1958). 
" P. Schofield, Phys. Rev. Letters 4, 239 (1960). 



200 



RESONANCE ABSORPTION OF NUCLEAR GAMMA RAYS 



1099 



here 



o- r r fko hV 
<r a (E)= — exp — — — 



4A l2k B T SMk B T 



x£exp[ 



•i-"l)--\t\] 
2h J 



XF, c (i,t)dTdt, (36) 



.(£) = 



ica ftV 



2£ s r 8Af* fl r 



<J exp t(«.*-«0 — -|*| 



XFS(T,t)dTdt, (37) 

where F, c {x,t) is the classical self -diffusion function 
and^=£— £ - 

We shall consider two simple cases : (i) in which the 
diffusive motion of an atom in the absorber is governed 
by the simple diffusion equation, and (ii) in which the 
atom jumps from one lattice site to another. Diffusion 
in normal liquids probably comes under case (i) whereas 
in solids it comes under case (ii). These two cases are 
considered here more as illustrations rather than to give 
a precise relationship between the resonance line shape 
and diffusive motions. 

Case (i) 

In this case F, c {x,t) is given by 

F/(r,/)=(4xZ)|/|)-*exp(-r 2 /4Z)|/|) I (38) 

where D is the diffusion coefficient. This function has 
the right limiting form for large t but not for / — » 0. 
One should rather use for F, c (T,t) the solution of 
Langevin's equation for the Brownian motion with p(t) 
as given by (13A). p{t) varies as P for t — » 0. We shall 
not use the latter form of p(t) since the change in the 
linewidth as a result of this refinement is negligible (see 
Singwi and Sjolander 15 in connection with neutron 
scattering). 

Let us suppose that the absorber is in the liquid state. 
The cross section for the absorption of a 7 ray of energy 
E is obtained by substituting (38) in (36) and after 
performing the integrations we have 

<r (£) = (aoT/4) exp (-2W a ) {T+2h?D)f 

[(£-£o) 2 +Kr+2^c 2 Z)) 2 ] ) (39) 

where in deducing (39) we have replaced 

expi-(E-E )/2k B T2 

by unity, since E—Ea<^2kBT in the resonance peak 
and where we have put h 2 ic i /SMkBT=2W a . 2W a is 



analogous to the Debye- Waller factor in the case of a 
solid. 

From (39) it is evident that the broadening Ae of the 
resonance line due to diffusion is given by 



Ae=2b<?D 
= 2Eo i D/kc*, 



(40) 



where £0 is the energy of the 7 ray and c is the velocity 
of fight. As an example let us take iron for which 
D~W~* cm 2 /sec in the molten state, and £ =14 kev 
(for Fe 57 ). Equation (40) gives Ae~6XlO~* ev which is 
several orders of magnitude greater than the natural 
width T. 

Experimentally what one measures is the self- 
absorption cross section 



<r(s)= 



i 



w e (E)a a (E+s)dE. 



Here w e (E) is the emission probability in the case of a 
solid, since the emitter is in the form of a solid. If we 
neglect the phonon part, the expression for w e (E) is 



r 1 

w,{E) =—e~* w ' . 

2t (£-£ ) 2 +r 2 /4 



(41) 



li K. S. Singwi and A. Sjolander, Phys. Rev. 119, 863 (1960). 



Using (39) and (41) in the expression for <r(s), we get 

o-oT T+hJD 
r(i)« — exn(-2W.-2W m ) . (42) 

2 5 2 +(r-r-^Z?) J 

Recently a cold-neutron scattering method has been 
used to measure the diffusive broadening of the "quasi- 
elastic" scattering in liquids. 16-18 This method, un- 
fortunately, suffers from the disadvantage of having a 
poor energy resolution. But if we were to study the 
diffusive broadening by Mossbauer technique, such a 
disadvantage does not exist since the natural width of 
the line is negligible compared to the diffusive broaden- 
ing. However, this method seems at present to be hardly 
practicable because of the smallness of a(s) compared 
to the other non-nuclear cross section such as the photo- 
electric effect. But under very favorable circumstances 
such that r/Ae is not too small this method could be 
used to investigate the shape of the resonance line and 
determine the diffusion coefficient from the measure- 
ment of line broadening. 

Case (ii) 

Let to be the mean time for which an atom stays on 
a given lattice site before jumping to a new lattice 
position. If we now assume that there is no correlation 
in motion between one jump and the next, it is possible 

11 B. N. Brockhouse, Phys. Rev. Letters 2, 287 (1959). 

17 D. J. Hughes, H. Palevsky, W. Kiev, and E. Tunkelo, Phys. 
Rev. Letters 3, 91 (1959). 

18 1. Pelah, W. L. Whittemore, and A. W. McReynolds, Phys. 
Rev. 113, 767 (1959). 



201 



1100 



K. S. SINGWI AND A. SJOLANDER 



to calculate F, c (t,1) or rather its Fourier transform as 
has been done earlier by Singwi and Sjolander 18 in con- 
nection with diffusive motions in water and cold- 
neutron scattering. The problem under consideration 
is in fact a special case of the more general formula of 
Singwi and Sjolander. The present case consists in 
taking ri — > 0, where n is the mean time for which the 
particle diffuses between its two oscillatory states and 
in taking the function h(r,l) to be independent of time. 
It is then easy to show by the use of formula (36) that 

* a (E)=— exp(-2P*g 

r+(2A/ro)(l-a) 

X- — : — — — , (43) 



where 



(£- £o) 2 +[r+ (2*/t ) (1 -a) J/i 
a*? f exp(tK-r)A(r)rfr, (44) 



h{t) gives the probability of finding the particle at the 
position r after a single jump, if the particle was at the 
origin before the jump. Using (43) and (41) we have for 
the self-absorption cross section 



<r r 

*(*)= — txp(-2W e 
2 



■2W a ) 



r+(*/ro)(l-«) 



^+Cr+(ft/ro)(l-a)? 
The diffusion coefficient D is denned by 



=— [r*h(r)di 
6to«/ 



(45) 



(46) 



From formula (43) it is clear that the broadening A« 
of the resonance peak due to diffusive motions is 

Ae=2ft/ro| 1- fexp(«e-r) h(t)dt\ (47) 

We thus see that the maximum value of the broadening 
is 2h/ro and the broadening depends on the angle be- 
tween the direction of motion of the diffusing atom and 
the direction of the 7-ray quantum. Consider a Fe 87 
nucleus sandwiched between two layers of a graphite 
lattice (it is possible to introduce iron atoms between 
the layer planes of a graphite single crystal). The Fe 87 
nucleus finds it hard to move in the direction of the 
c-axis but can diffuse with ease in the basal plane. If 
the 7 ray from the emitter falls on the absorber parallel 
to the c axis and the counter is also pointing along the 
c axis, the diffusive broadening Ae in this case will be 
negligible and the resonance line will have its natural 
width. If we now rotate the absorber relative to the 
direction of the incident 7 ray, the diffusive broadening 



should increase. At the same time the magnitude of the 
resonance absorption would decrease due to the ani- 
sotropy of the Debye- Waller factor (our formulas can 
easily be extended for an anisotropic solid). 

Let us consider self-diffusion in iron. The emitter is 
at low temperature and the resonance absorption is 
studied as a function of the velocity of the emitter for 
various temperatures of the absorber. At ordinary tem- 
peratures, the self-diffusion in iron is so small that the 
line broadening due to diffusive motion is negligible 
compared to the natural width T. For example even 
at 760°C, the diffusion coefficient is only 1.5X10-" 
cm 2 /sec, which would give a value of 6X 10^ 6 sec for r ; 
since r^iP/6D t I being the interatomic spacing. This 
would correspond to a broadening (A6) mAX ~2X 10 -11 ev, 
whereas r=4.6Xl0~ 9 ev. However, at higher tempera- 
tures, say 1000°C and above, it should be possible to 
detect the diffusive broadening by a Mossbauer-type 
experiment. At such high temperatures the Debye- 
Waller factor (T 2W ° (since 2W,«2W„) would no doubt 
decrease but it is still not too small (~c -a ) as to pre- 
clude the possibility of observing the resonance effect. 

It has been reported 19 that the rate of self-diffusion 
in iron at 757°C, under plastic deformation, increases 
by as much as a factor of thousand. 20 And if it is true, 
the line broadening due to diffusion would now be 
nearly 2X 10"~* ev which is greater than the natural line- 
width and it might, therefore, be possible to detect it 
by a Mossbauer-type experiment. It would be valuable 
to perform such an experiment in view of the fact that 
there exists a controversy between different experi- 
mental workers regarding the enhancement of the self 
diffusion and the range of temperature for which it is 
significant. Besides a Mossbauer-type experiment is 
different from the usual diffusion experiments using 
tracers and should, therefore, provide an independent 
check. One could also study the impurity diffusion of 
iron in other metals like copper and silver. Such an 
experiment would give a direct measure of the mean 
time to and its temperature variation. 

APPENDIX 

The function G,(t,t) as defined by Eq. (3) could only 
in the classical limit be interpreted as the self-diffusion 
function. For small times when the particle under con- 
sideration has moved only a distance of the order of the 
de Broglie wavelength, quantum effects are important 
and G,{t,t) is complex. In fact, according to Van Hove 8 
it is only the real part of G,{t,t) which has the above 
physical interpretation. It is, however, possible to ex- 
press the emission or the absorption probability in 
terms of the real part of G.(T,t). 



u N. Ujiive, B. Averbach, M. Cohen, and V. Griffiths, Acta 
Met. 6, 68 (1958). 

M For a general discussion see the review article by D. Lazarus, 
Solid State Physics, edited by F. Seitz and D. Turnbull (Academic 
Press, Inc., New York, 1960), Vol. 8, p. 71. 



202 



RESONANCE ABSORPTION OF NUCLEAR GAMMA RAYS 



1101 



Now 



1 r- 

— I exp(—*u/)<exp[— tie- R(0)] expftie- R(/)])tA 

= En.»og»oK»|exp(iK-R)|no}| 2 

XS|>-(e»- €„„)/*], (1A) 

which follows from the definition of ( • • • )? and g nn o 
= exp(—E n0 /k B T)/Y, n exp(—E n /k B T). Further 

1 /•» 

— I exp(-ia>/)<exp[-iie- R(/)] expftie- R(0)]) r <# 
2irJ- n 

=E« ! »«g»o ( K»|exp(tK-R)|«0)p8[a)+(« n -€ B o)/^] 

= L»,no^o|<«0|exp(iK-R)|»)|25[a,-(e„-6»o)A] 

= e-**"> L».»og»o|<»| exp(iK- R) | »0>| 2 

X*[«- (•»-•,*)/*] 
1 /•- 
=e -^ft<- — I exp(— tarf) 
2ir«L« 

X(exp[-iie- R(0)] expftie- R(0]>j<ft. (2A) 

The last step in (2A) follows from (1A). The relation 
g n =exp[— /?(«»— e„o)]gno has been used in the second 
step in (2A) ; 0= l/*j,7\ 

Introducing the real part of G,(r,t), which is defined 
by 

Re[G.(r,0]= (2tt)- 3 f exp(-«e- r) 

Xi{<exp[-«K- R(0)] expftie- R(/)] 

+exp[- »k- R(0] expftie- R(0)]> r }rfK, (3A) 

and using (1A) and (2 A) in Eq. (5) of the text we get 
the following expression for the absorption cross section : 



<r a {E)- 



a T exp(/3fto>/2) 
4ft coshG8&o/2) 



X J expft (k- T-af) - (r/2A) | / 1 ] 

XRe[G,(r,fl]</r<ft f (4A) 
considering that T<&k B T. 

Recently Schofield 14 has also suggested in connection 
with neutron scattering that G,(r, t+ih/2k B T) rather 
than G,(x,t) should be considered as a self-diffusion 
function. He points out that if, for instance, G,(r,t) is 
replaced by its classical equivalent obtained from the 
simple diffusion equation, as suggested by Vineyard, 6 
the scattering cross section will not satisfy the condition 
of detailed balance. If, however, G s (r, t+ih/2k B T) is 
replaced by the classical self-diffusion function the 
principle of detailed balance will be satisfied. The same 
is also true if we replace Re[G,(r,t)~] in (4A) by its 
classical equivalent. 



Schofield's result is easily obtained by noting that 

1 r » 

— I exp(-t«0(exp[-tie- R(0)] expftie- R(0]>t* 
2-n-J^ 

=L».»og»o|<»| exp(«- R) |nO)| 2 S|>- (e»-«„o)/ft] 

= exp(/3fta J /2)Ln,no(^„o) i Kn|exp(iK-R)|«0>| 2 

X«[>-(«»- «„„)/*], (5A) 

and introducing a function 

F.(r,0=(27r)- 3 Jexp(-iie-r) 

X{Z».»o(g»f»o)»<«0|exp[-iK-R(0)]|») 

X<n|exppK-R(0]|nO»rfie. (6A) 

We then have for the absorption cross section 

<roT r 

a.(£)= — expC8&o/2) I expp(ieT- «fl-(r/2ft)|*|] 
4ft J 

XF,(r,t)drdt. (7 A) 

F s (r,t) is real and is an even function of t and thus the 
integral in (7A) is an even function of co. It is easily 
shown by using the definition (3) of G,{r,t) that F,{x,t) 
= G,(i,t+ih/2k B T). If one uses the classical self- 
diffusion function instead of F,(t,t) in (7A) or for 
Re[G„(r,0] in (4A), the two expressions are identical 
to the first order in tiw/k B T. 

If we make the transformation as suggested by 
Schofield, we have 



F.(r,/)=[2Tp(0]-»exp[-fV2p(0], 



(8A) 



where 



P (0 = &/4Mk B T+ (k B T/M)P, (9A) 

for a free gas, and 

P (/) = (h*/M ) tanh (z/U B T)dz 

J 2 



+ (ft 2 /M)f 



/(a) l-cos(2//ft) 
z smh(z/2k B T) 



dz, (10A) 



for a solid. Equations (9A) and (10A) follow from Eqs. 
(11) and (17) of the text, respectively. 

We notice from (9A) that even at /=0, the particle 
is distributed over a finite region. The finite extension 
is given by the first term in (9A) and is consistent with 
Heisenberg's uncertainty principle for a particle with 
mean velocity (k B T/M)l. The real part of G,(r,t) on the 
other hand, goes over to a S function around the origin 
at / = 0. It, therefore, appears that F,(r,t) as given by 
(8A) is more directly connected with self-diffusion. 

Schofield's suggestion, in the case of a liquid, is to 
replace F,(T,t) in the first approximation by a classical 



203 



1102 



K. S. SINGWI AND A. SJOLANDER 



self-diffusion function obtained, for instance, from 
Langevin's equation. In that case p(t) will go to zero 
as / 2 for small times and will approach 2D\t\ for large 
times. If we, however, add a constant to p(t) corre- 
sponding to a finite extension of the probability cloud 
at /=0, the resulting formula for the cross section will 
be valid to some extent also for large momentum 
transfers. 

For a liquid it seems reasonable to take the same con- 
stant as that for a gas, since we know that for large 
momentum transfers corresponding to small times the 
scattering cross section approximately goes over to a 
free gas formula. Adding of this constant to p(t) will 
simply amount to multiplying the right-hand side of 
(7A) by exp(-n 2 K 2 /SMk B T). In the case of neutron 
scattering by liquids this factor is often nearly equal to 
unity except for large incident neutron energy, whereas 
in the case of 7-ray resonance absorption it could be 
quite small depending on the recoil energy of the 
nucleus. 

As a result of the foregoing discussion it seems plau- 
sible to write (7A) in the form 

a a (E) =— exp(ho/2k B T-hV/BMk B T) 
4# 

X f expp(K-r-u0-(r/2ft)|f|] 

XF. e (r,t)drdt, (11 A) 

and similarly, 



w.(E) = — exp(-ha>/2k B T-hW/SMk B T) 

2wh 

X J expP(KT-«0-(r/2fc)|/|] 

XFS(t,t)didt, (12A) 

where for F, e (r,t) we take the expression (8A) with 
p(t) as given from Langevin's equation, and is 21 

p(0= (2Zy/3')D3'/-l+exp(-/3'/)]. (13A) 

The characteristic time 1//3' is given by 

P'=k B T/DM, (14A) 

B See for instance S. Chandrasekhar, Revs. Modern Phys. 15, 
1 (1943). 



D being the diffusion coefficient. Of course, this is 
possible only if the diffusion can really be described by 
Langevin's equation. 

Note added in proof. Recent measurements of the 
specific heat of indium by Bryant and Keesom [Phys. 
Rev. Letters 4, 460 (1959)] and of niobium by Broose 
et al. [Phys. Rev. Letters 5, 246 (I960)] both in the 
superconducting and normal phases seem to show that 
the lattice part of the specific heat is different in the two 
phases, thus indicating that perhaps the phonon spec- 
trum in the two phases is not the same. Broose etal., in 
an attempt to explain their measurements on niobium 
have suggested in the superconducting phase an altered 
value of the Debye temperature as 243°K, which value 
in the normal state they arrive at is 231°K. Thus, there 
is a change of five percent in the value of Bo- 

Here we wish to suggest an alternative and perhaps 
more direct experiment to decide whether there is any 
appreciable change in the value of 6 D in going from the 
normal to the superconducting phase. The experiment 
consists in studying the intensity of the Mossbauer line 
both in the normal and superconducting phases. The 
choice for such an experiment is very severely limited to 
only a few isotopes. The intensity of the Mossbauer line 
is determined by the Debye- Waller factor e~ iw , and in 
the limit T<K8 D , 4w is equal to 3R/k B d D , where R is the 
recoil energy. In order to have an appreciable change 
in the intensity of resonance absorption for a very small 
change in the value of 6 D , one would demand a large 
value of 4a> ; i.e., a large recoil energy and a small do- 
Re 187 is such an example. It becomes superconducting 
and the values of R and d D are, respectively, 0.051 ev 
and 417°K. An almost trivial calculation will show that 
a ten percent increase in the value of Od will give a 35% 
increase in the intensity of the resonance absorption, 
which should be easy to observe. Other isotopes which 
one could study are Ta 181 and Hf m . 

The authors have been informed privately by Meyer- 
Schuetzmeister and Hanna that their very preliminary 
experiment on the Mossbauer effect in Sn, both in the 
normal and superconducting phases, indicates that there 
is an increase of a few percent in the intensity of reso- 
nance absorption in the latter phase. A five percent 
increase in the value of do would, in this case, give nearly 
three percent increase in the intensity. 



204 



LE JOURNAL DE PHYSIQUE ET LE RADIUM 



TOME 22, MAI 1961, PAGE 303. 



DIFFUSION DES PHOTONS SUB LES ATOMES ET LES NOYAUX DANS LES CRISTAUX 

Par C. TZARA, 

Section de Physique Nucleaire a Moyenne Energie, C. E. N., Saclay. 

R6sum6. — On etudie la diffusion des photons par les atomes et les noyaux lies dans un solide 
avec ou sans cession de phonons. Selon la largeur de l'etat excite (atomique ou nucleaire), relati- 
vement au spectre de vibration du cristal, la proportion de diffusion « sans recul » varie entre deux 
limites, l'une etant le facteur de Debye-Waller, l'autre le carre du facteur de Mossbauer-Lamb. 
L'interference entre les processus atomique et nucleaire est examinee. En fin remission sans recul 
d'un photon nucleaire a la suite d'une cascade de transitions est calcnlee sans faire appel a un 
phenomene de rearrangement des etats cristallins. 

Abstract. — Photon 'scattering by atoms and nuclei in solids is investigated. Depending on 
the excited (nuclear or atomic) state width compared to the crystal vibration spectrum, the pro- 
portion of recoilless scattering varies between two limits : the Debye-Waller factor on one side, 
the square of the M6ssbauer-Lamb on the other. Interference between the two processes is 
examined. Finally the recoilless emission of a nuclear photon following a sequence of transitions 
is calculated without requiring a rearrangement phenomenon in the solid. 



L'interaction des photons avec les atomes ou les 
noyaux lies dans un solide peut se produire sans 
cession de phonons au cristal. La diffraction des 
rayons X est un exemple de ce phenomene. Recem- 
ment Mossbauer a decouvert remission et l'absorp- 
tion sans recul de photons par les noyaux dans un 
solide [1]. Dans le present travail, nous examinons 
entre autres les diffusions atomique et nucleaire 
dans les solides et leur interference eventuelle en 
utilisant la theorie de Lamb [2]. 

Le cristal est compose d'atomes de masse M, 
a Z electrons. Les etats et energies propres du sys- 
teme electronique sont <Jn et E{, ceux du noyaux 
sont 9j et W\. L'etat du cristal est defini par l'en- 
semble j a. B ) des nombres d'occupation dans chaque 
mode de vibration s. L' energie propre de l'etat { a, } 



est E\ a, ) = S a, hta,. A l'etat d'equilibre, a 

M 

temperature T, le poids statistique de chaque 
etat | «, } est g { a, }. Dans le cas ou une espece 
atomique est inseree dans un reseau d'un autre 
constituant, ce ne sont pas les degres de liberte de 
l'ensemble qui sont a considerer, mais ceux des 
atomes interesses. 

1. Diffusion Rayleigh atomique. — La section 
efficace de diffusion d'un photon d'energie k t impul- 
sion K et polarisation € en un photon d'energie k', 
impulsion K' et polarisation e', le cristal 6tant 
initialement et finalement dans les etats purs { a, } 
et { p, j, l'atome etant laiss6 dans l'etat initial, est, 
en approximation non relativiste : 



^f ( j «.}{ p.} JHTm') = I < { P.}|e^«/« | j a , | > < fc | ^,W«|fc > €€ ' 

+ 1 S <\t.]\e- iK '«l*\[n,]> <|n.}le'*"/»l{«.j> < ^ \I> e~ iK ' r ^ p'Mn >< j>n\l> e ^f" Pi U > 
m {».}• ■ E P — E - k + £|» # | — £| a< | + if 



£„ — E + k' + fj^j — ^j^j 



+ iY 



X [*fm*)*k'lk 
ici q = K — K' transfert d'impulsion 

k' + ei*\ = k + m,\ 



(i) 



Le premier terme d6crit la diffusion Thomson par 
les Z electrons arranges selon la configuration tj» . 
Le facteur de forme : 



ZF[q) = < +„|Se^/*|+,> 



u : coordonnee du centre de gravite de l'atome 
consider^ dans le cristal ; 

rj : coordonnees des electrons atomiques dans (dans le cas d'un atome sphenque) suffit genera- 
un systeme lie a l'atome. lement a rendre compte de la diffusion cohe>ente si 



205 



JOURNAL DE PHYSIQUE 



N°5 



l'energie du photon est superieure aux energies des 
discontinuites d' absorption E n — E les plus, fortes. 
Nous ne nous limiterons pas pour l'instant a ce 
terme. 

Si k est tres different de E n — E 0) E« t — E^ 
peut §tre neglige dans les denominateurs des termes 
dispersifs du premier type ; si k ~ E % ■ — E , cette 
approximation reste valable car la partie imaginaire 

Y c± k r„, la largeur de l'etat ty n est, pour les 
transitions les plus fortes, tres superieure a 
Ei i — Ei a i cz kQ, etant la temperature de 

Debye du solide. 

Dans les termes dispersifs du deuxieme type, 
l'approximation est evidemment meilleure. La som- 
mation 2 s'effectue alors aisement puisque : 



S || B ,}>< I ».)|-1 



et il vient : 

dQ 



£f{|a.}{M)KK'ec') 



= |<|p.||eW|j«.}>| 



\ZF[q) €.€' + M+ + ilf-| 2 (e*lmcy^. (2) 

Dans ce cas, le plus repandu, la section efficace 
dans un cristal est simplement le produit d'un fac- 
teur o cristallin » par la section efficace atomique. 
Si la section efficace varie rapidement dans des 
intervalles d'energie de l'ordre de kQ, cette factori- 
sation n'a plus lieu. C'est le cas des resonances de 
reaction (rey) [2], avec, ici, la complication supple- 
mentaire des termes dispersifs. 

Deux quantites facilement observables sont de- 
duites de (1) : la section efficace de diffusion totale, 
integree sur tous les etats finals possibles du cris- 
tal, -r^ (KK'ee') et la section efficace de diffusion 

sans recul, -tt? (KK'et'). 

En nous placant dans les conditions de validite 
de (2) : 

do ! = JS 1 *l*li<IMi«w«i|«.|>i'*?. 



dQ 



WN 



d^B 



Si nous negligeons la variation de —? en fonc- 

d 12 

tion de k\ ' = k + E> \ — Ei & i en utilisant : 
2||f.!><|P.)l = l et 2 f|«}-l 



nous obtenons evidemment : 

dok/dO = da E /dO 
et : 

do?/dfl= S g(« t j|<{|a.}|e i « tt /' l |{a.)>| 2 daE/dn. 
M 
Pour un cristal isotrope, on trouve le facteur de 
Debye-Waller par un calcul immediat calqu6 sur 
celui de Lamb : 



d_a E 
d 



f b j 3 q- l r fi kv du 1 ) 



d«E 
dl2 



(3) 



X = IT. 



2. Diffusion nucleaire resonnante. — La diffu- 
sion resonnante est un phenomene a un seul quan- 
tum ou encore coherent, lorsque l'etat du diffuseur 
pendant le processus n'est pas observe [4]. C'est le 
cas de toutes les diffusions et reactions, ou le temps 
de vie de l'etat excite etant tres court, le faisceau 
incident (la perturbation) et le faisceau diffuse 
sont pratiquement etablis en permanence. Gepen- 
dant les experiences ou un delai est introduit entre 
l'onde incidente et l'observation d'une particule 
reemise [5] detruisent cette coherence. Notons aussi 
que les transformations a ou (3 et les cascades de y 
des noyaux peuvent etre envisagees sous cet aspect ; 
la coherence peut etre detruite simplement en 
isolant l'etat intermediaire s'il a une vie assez 
longue ou bien par une interaction exterieure telle 
que le couplaga entre le moment quadrupolaire de 
l'etat intermediaire et le champ cristallin. 

Puisque la coherence et l'absence d'information 
sur l'etat du diffuseur pendant la diffusion sont 
liees, on s'attendrait a ce que la proportion de 
diffusion sans recul soit donnee par 



exp 



2 2MKG 



W* 



ou figure le transfert global d'impulsion q et non 
par 

K* 1 



exp { — 3 



2Mk& 



F[x) = /» 



ou figure le transfert d'impulsion lors de i'absorp- 
tion ou de remission. En realite le r6sultat varie 
entre ces deux extremes selon la largeur de l'etat 
excite. En effet, en appelant 

MIKK'et') 
W l —W —k + tT/2 

l'amplitude de diffusion resonnante, la section effi- 
cace de diffusion sans recul est : 

|2 <{«||e«'«/»|{»,}> <j 



^ Ka /*l{«.|>| 2 



|M(KK'€€')|», (4) 



206 

No 5 DIFFUSION DES PHOTONS 305 

Les deux cas limites sont : 1° Largeur de l'etat type de Bragg ne peut avoir lieu que si le cristal 

excite <p x grande devant A;0 ; deja examine plus est laisse dans l'etat initial). Dans le premier cas : 

haut : y. 

i« ( ..j - B |..|i « \w, - w.- * + i-A ra = N s ! - ! ^ l( - l ' - ! **'"'' 

d<JB j 2 da r 

et on obtient : = ? ^q + / ^ 

da?/dO = 9 da r /da + S g{ a, j|< | «. )|.W»|j a. ) >< j a, }| 

2° Largeur T < k&. Alors |Wi — W - k\ ~ V. 

Les transitions par des etats intermediaires du e lK ' u ^\\ «,}><{ a, ]| e — lKu l h \ 
cristal { n s } ^ { a s ) ne contribueroht pas si 

Ei n i - £| a i » T. 11 ne pourrait y avoir de con- j «, j > — _ £* S _^*_ - r/2 + C.Cj x e*//«c- a 

tribution que de la part de transitions aunou , . ... , , 

quelques phonons tres mous : *«.. -* 0. Mais leur au P remier ordre les elements de matrice : 

densite diV(o))/ato varie en to 2 /© 3 ; on peut done < < u e ipu/h,i aj > > 

les negliger, d'autant mieux que la temperature de ' * ' ' * ' 

Debye est elevee. La somme dans (5) se reduit au ^ if < li + 1 pf ,„ N , J > 

terme { n 3 j = { a 3 ), la partie imaginaire y est ~s-i l 2 2M7Vsg>» * '\ a ' 

egale a 172. On a done : x , , , 

sont reels, done : 

da?/dQ = S g{ «,}|<{ a. ||e- iKM / fl |{ « s j > | 2 da _ doR da_ r 

[ a *j dft~ 9 dft + ' dQ 

|<|«.}|e**/«||«.}>|»d« r /dn + 9^ L_^_, T , a + C.C.l 



w x _ w — k — ir/2 



K l 1 „, Vl > da r „ da 



exn 3 — — FIX) — = / 2 — -^k ^* est reel. Le terme d'interference varie done 



2M/r0 v ' dQ 



Dans le cas intermediate d'une largeur nucleaire 



avec l'energie comme 

W, — W, — k 



assez importante, le resultat est plus complique, l - Wl w ° k ^ + r2 /* 

car la factorisation en termes dependant du cristal c . , „ „t,„x„„ b ■ ■ j „+„„„„„• + ., „„„„.. 

Pt du novan nP sp fait nlns Sl les P hotons incidents proviennent d une source 

et au noyau ne se iait pms. dg mgme nature 6mettant sans recul? integration 

1 
3. Interference entre diffusion atomique et diffu- sur le s P eGtre ^y^—Wo — k) 2 + r/4 fait dispa " 
sion nucleaire r&onnante. — Dans le cas le plus raitre le terme d'interference. Celui-ci ne subsiste 
frequent, la diffusion Rayleigh est monotone, et la que si la raie incidente est dissymetrique par rap- 
diffusion nucleaire a lieu sur un niveau tres etroit. port a W x • — W , par exemple si la source est ani- 
Nous avons dans ces conditions : mee d'une vitesse qui decale l'energie de (v/c)k. 

La diffusion totale a pour section efficace, en 

— (i a, } I $, } KK'ce') admettant 1' approximation k' = k : 

= |<| P .!|e^|( as |>^(^) 1 %B(K K 'ec') SrNW f l*iS^)l-> 

+ < { P. ||e-*'«/*|[ «,)><[«. )le^«/«,j „ ) > - |£ + / Jg 

M(KK'zt') |« 2(Wl — W -k) ^ 



a quoi s'applique la meme discussion que pour 
da 2 /dO. 



W x — w —k + iF/2 

ou : 

AR(HK'ee') = ZF(q) e.e'+ ••• 

Conclusion. ■ — La diffusion atomique et la diffu- 

L'interference entre les deux amplitudes a lieu sion r&sonnante nucleaire sont des processus cohe- 

quel que soit l'etat final du cristal (l'interference du rents. Les facteurs de Debye-Waller pour ces deux 



207 



JOURNAL DE PHYSIQUE 



N°5 



processus dependent essentiellement de la largeur 
des niveaux vis-a-vis du spectre de vibration dti 
cristal. Pour des raies tres etroites, la diffusion pro- 
cede par un etat intermediate du cristal bien defini 
identique a l'etat initial. Pour des raies tres larges, 
ou une section efficace monotone, l'etat inter me- 
diaire du cristal n'est pas defini : le processus passe 
par tous les etats intermediaires possibles du cristal. 
En d'autres termes, dans le cas de raies tres 
etroites, le temps caracteristique de diffusion est 
long, la configuration des positions de l'atome dans 



le cristal pendant la diffusion est moyennee, ce qui 
correspond au melange statistique d' etats a l'6qui- 
libre. Pour des raies larges, ou des temps caracte- 
ristiques tres courts, la moyenne n'a pas le temps 
de se faire, la diffusion a lieu sur une configuration 
instantanee. Notons que le premier terme dans la 
diffusion Rayleigh decrivant la diffusion Thomson, 
comporte une sommation implicite sur des 6tats 
intermediaires. En effet l'expression rigoureuse de 
la diffusion Thomson fait apparaitre des termes dis- 
persifs de la forme : 



S < I P. || e- l ' K '"/«| { n, J >< \ n, j| e iKM /*|{ a ]>< pj.il>, > < Vi\aA\v > 

lim. < ) — — 

fc-o \ n s\ v i ± Vm 2 c 4 + c 2 p' 2 — mc 2 ± k + EisJ — £(» j 



les etats intermediaires s'ont a energie positive ou 
negative et l'ordre d'absorption et d'emission est 
inverse. Lorsque k = 0, les termes relatifs aux etats 
d' energie positive s'annulent.Seuls les etats d'6ner- 
gie negative contribuent, et leurs denominateurs 
sont de la forme 



— Vm* c 2 + c 2 p' 2 — mc 2 + EU\ — Ei*A 

-+ — 2mc 2 + £UI — £(«! 

Ei n > ■ — Ei^ j peut etre neglige, d'ou la possibilite 

de sommer sur les etats intermediaires { n 3 ]. 

L'interference entre la diffusion resonnante et la 
diffusion atomique est a rapprocher de Tinterfe- 
rence entre photoelectrons et electrons de conver- 
sion [6]. 



L'auteur tient a remercier MM. Abragam, Bar- 
loutaud, Cotton, Picou pour les discussions qu'il a 
eues avec eux. 



APPENDICE 

Emission d'une cascade de deux photons. — Le 

noyau est initialement dans l'etat excite d'ener- 
gie W 2 , de largeur infiniment fine, et en equilibre 
thermique dans le reseau. II emet un photon k 2 , 
K 2 (ou un electron de conversion E z , p 2 ) par tran- 
sition a l'etat y x d'energie W de largeur T puis un 
photon au fondamental. Le probleme est de cal- 
culer la probability d'emission sans recul du pho- 
ton k v Elle est proportionnelle a : 



g\ *s 



Z<\V,\\e iK L u I h \\n s ]> < 



J K ,«/*\\ a, I > 



W 1 — W 2 + k 2 + £j» s | - £ja,} + iy 

h + k 1 + 2?|f» i = W , + Elai. 



(1) 



(2) 



L'emission du photon k x sans recul signifie que 
l'etat du cristal ne change pas au cours de la deu- 
xieme transition : { rig \ = { (S„ }. En meme temps 
nous nous placons dans l'hypothese que l'etat cris- 



tallin intermediaire subsiste plus longtemps que 
%/F autrement dit qu'il n'y a pas de rearrangement 
spontane pendant le temps de vie de l'etat inter- 
mediaire. Alors (1) devient, avec l'aide de (2) 



«l*l 



< | P. ]\e iK ^\\ p. | >< j P s }|e«««/*|{ a, \ > | 2 



[W x — ki) 2 + T 2 /4 



208 



N°5 



DIFFUSION DES PHOTONS 



307 



La raie k x est done emise avec sa largeur natu- 
relle. Le terme cristallin s'evalue comme suit : 



n i< 



M ?.)H-*5.(2iV, + l)||p.}>| 

rll •) *- x 

|< | P« jll — ^ (2^* + 1) + **{«. + «. + )l| *. 

en posant 



g{ *. 



3tf 

n 



Kl c(2MiV*w,)-'/s 
-Ar|,[2(a, + 1) + 1] } a, A«, 



+ { 1 — *«,(2a, + 1) } | 1 — A| s (2a, + 1) } 

+ j 1 — *f.[2(a, + 1) + 1] | («, + 1) *t,] 

au premier ordre en Af, il ne reste que 

S{k|n[l — Wa, + 1)] 

N 



(3) 



(3) 



qui est Pexpression du facteur d'emission sans 
recul d'un photon k x egal, pour un cristal de Debye 
a : 



ex P — o51S 



2 2Mk@ 



F(X) 



Geci demontre le fait experimental reconnu que 
remission d'un photon Mbssbauer a lieu indepen- 
damment de 1'histoire anterieure du systeme, sans 
qu'il soit utile pour l'expliquer de faire appel a un 
rearrangement du cristal apres les emissions prece- 
dentes. Les etats intermediates du cristal se corn- 
portent comme un etat d'equilibre pratiquement 
identique a l'etat d'equilibre initial g j oc, {. 

L'expUcation par le rearrangement cristallin se 
revele necessaire si le recul X|/23f est considerable 
et d6truit localement le reseau. 



Manuscrit recu le 5 Janvier 1961. 



REFERENCES 



[1] Mossbauer (R. L.), Z. Physik, 1958, 151, 124. 
[2] Lamb (W. E.), Phys. Rev., 1939, 55, 190. 
[4] Heitler (W.), Quantum Theory of radiation, Oxford, 
University Press. 



[5] Holland (R. E.), Lynch (F. J.), Perlow (G. J.) et 
Hanna (S. S.), Phys. Rev., Letters, 1960, 4, 181. 

[6] Cf. LiPKiN.dans "Mossbauer Effect ■>, ed. de 1'Univer- 
site d'Dlinois, TN, 60.698. 



209 



SEANCE DU 11 AVRIL 1960. i^o5 

PHYSIQUE NUGLEAIRE. — Diffusion resonnante du rayonnement v de 
23,8 keV de il9 Sn* emis sans recul. Note (*) de MM. Roland Barloutaud, 
Jean-Loup Picou et Ciiristopue Tzara, presentee par M. Francis Perrin. 

Nousavons observe les photons reemis par 119 Sn apres absorption resonnante de 
photons 6mis sans recul par l1 "Sn*. La valeur du facteur de Debye-Waller, a 
environ 90 K, mesur^e par cette m£thode, est en bon accord avec celle mesur^e 
par transmission. 

De recentes experiences ont mis en evidence l'emission et l'absorption 
sans recul de photons par des noyaux lies dans un cristal. Cet effet decouvert 
par Mossbauer (*) a ete jusqu'a maintenant observe au moyen d'experiences 
de transmission a differentes temperatures. Nous avons etudie les rayon- 
nements y reemis apres une absorption resonnante sans recul. Une telle 
experience est en principe plus sensible qu'une experience de transmission; 
en effet, les photons emis sans recul ne sont diffuses notablement que par 
interaction avec les electrons atomiques, les sections efhcaces correspon- 
dantes etant generalement beaucoup plus petites que celles de la diffusion 
par resonance nucleaire. 

Les meilleures conditions d'observation de ce processus sont obtenues 
lorsque : 

i° La diffusion non resonnante est peu intense, c'est-a-dire aux grands 
angles ou les divers facteurs de forme de la diffusion atomique sont petits. 

2 La section efficace de diffusion resonnante, proportionnelle a i/E 2 (i+ a ) 
est elevee (E etant l'energie de la transition et a le coefficient de conver- 
sion). 

3° Les facteurs de Debye-Waller, donnant la proportion des processus 
sans recul, sont grands. 

Cette experience a ete faite avec le rayonnement y de 23,8 keV de 
iiog n * pj dont le schema de disintegration est represente sur la figure 1. 
Ce noyau remplit assez bien les conditions ci-dessus et presente en outre 
l'avantage den'emettre, a part le rayonnement y interessant, que les raies X 
de l'etain. Un ecran de palladium dont la discontinuity K est situee 
a 24,36 keV, absorbe selectivement les X K de l'etain sans attenuer for- 
tement le rayonnement y. 

Le dispositif experimental est schematise sur la figure 2. Les photons 
sont diffuses a 70 ± 5° par un cylindre d'etain enrichi en ii9 Sn (71,5 %) 
maintenu a l'interieur d'un cylindre en mylar de 0,07 mm d'epaisseur. 
Un ecran de plomb masque completement la source vue du cristal. L' en- 
semble source diffuseur se trouve a l'interieur d'une boite en polyethylene 
expanse et peut etre porte a une temperature voisine de celle de l'azote 
liquide. 

Les mesures d'intensite des photons diffuses ont ete effectuees a la 
temperature ambiante et a environ 90 K. La diffusion due a la matiere 



210 

2706 ACADEMIE DES SCIENCES. 

environnante a ete determinee par une experience temoin faite sans diffu- 
seur de 119 Sn. Les taux de comptage obtenus, corriges de l'effet d'emission 
des X K du palladium, sont respectivement : 



N 90 »r = i3,9 ± 1 par minute, 
N :J oo<>r= 4,65 ±0,7 par minute; 



d'ou 



N 3n o 



o,5. 



Ces intensites peuvent s'exprimer en fonction des sections efficaces 
differentielles de diffusion resonnante et non resonnante (da/dco) r et 



11 


T. 270 j 




89 UaV 


■ I 




£ grand 




♦ 3 


T. 1,85 10* 3 


i 


- 2J8 WoV 


I 




M. «.f. 


6,3 10> 


1 


' 


1 







Fig. 1. 



1. Soure* W 5n (SO ma/em') 

2. Dlffu»aur w 5n (5mg/em«) 
2>. Ecran da plamb 

k. Ccron de R* (120 mg/o^ 1 ) 

5. 5clnrllloraur da Nat (Tt) 

Fig. 2. 



(da/d(i)) n . T et de la proportion f L du rayonnement y emis ou absorbe sans 
recul a la temperature t en supposant que / 300 est negligeable : 

N 90 ~(i — f 90 )nl -^ J a n . r .+ (/«« — £90) S, 

m est le nombre d'atomes d'etain par centimetre carre du diffuseur a 
l'incidence 35°, £ 9 o est l'autoabsorption resonnante de la source a 90 K. 
S est la probability pour qu'un photon sans recul soit diffuse par reso- 
nance dans la direction du cristal : 



s=M^-* > + (-^)<-->]( 



ctto J,. 



t etant la transmission des photons sans recul a travers l'ecran a l'inci- 
dence de 35°; a r et a nr sont les absorptions electroniques moyennes des 
photons diffuses dans les processus resonnants et non resonnants. 

La section efficace des processus non resonnants (do/dw) B .r. a ete deter- 
minee a l'aide des facteurs de formes donnes dans ('); elle est egale a 



211 



SEANCE DU 11 AVRIL 1960. 2707 

19 barns/srad. La section eflicace resonnante {dajd<si) t a ete calculee en 
supposant que la distribution angulaire des photons correspondait a une 
transition dipolaire magnetique pure et n'etait pas attenuee par inter- 
action quadrupolaire, une recente mesure (*) donnant une limite supe- 
rieure de io~ 8 eV a Q d 2 Y/dz 2 . Elle est egale a 6 000 barns/srad. On deduit 
des resultats experimentaux : 

/ 00 =o,3o± 0,07. 

Cette valeur est en bon accord avec celle obtenue par des mesures de 
transmission (/ 90 = o,32 ± 0,01 5) (*). 

II est possible d'augmenter le rapport N 9 o/N 3 oo et d'ameliorer ainsi la 
precision en operant avec un diffuseur plus mince (/^ 1 mg) et un angle 
de diffusion plus grand de facon a rendre pratiquement negligeable la 
diffusion non resonnante. 

Cette methode presente d'autres aspects interessants. La distribution 
angulaire des photons diffuses est une source d'information sur le carac- 
tere de la transition et l'importance de l'interaction quadrupolaire; de 
plus F etude des interferences entre les processus sans recul resonnants et 
non resonnants ( 5 ), ( 6 ) peut apporter une verification de la coherence dans 
la diffusion resonnante. Enfin, la raie de photons sans recul absorbes 
et reemis sans perte d'energie est sensiblement plus etroite que la raie 
emise initialement par la source, la distribution en energie est par exemple, 
pour une source et un diffuseur minces, de la forme i/[(E — E ) 2 + F 2 /4] 2 > 
conduisant a une raie de largeur a mi-hauteur 0,64 fois celle de la raie des 
photons avant diffusion. 

(*) Seance du 4 avril i960. 

(0 R. L. Mossbauer, Z. Physik, 151, 1958, p. 124. 

( 2 ) R. Barloutaud, E. Cotton, J.-L. Picou et J. Quidort, Comptes rendus, 250, 
i960, p. 319. 

(') A. K. Compton et S. K. Allison, X-rays and experiment, Van Nostrand, i e €d. 
p. 781. 

(*) Sous presse. 

( 8 ) A. Kastler, Comptes rendus, 250, i960, p. 509. 

(°) G. Tzara et R. Barloutaud (sous presse). 

(Centre d'Etudes nucliaires de Saclay.) 



212 



Volume 4, Number 11 



PHYSICAL REVIEW LETTERS 



June 1, I960 



ZEEMAN EFFECT IN THE RECOILLESS y-RAY RESONANCE OF Zn OT t 

P. P. Craig, D. E. Nagle, and D. R. F. Cochran 

Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 
(Received May 4, 1960) 



Recoil-free resonance emission and absorption 
(Mossbauer effect) 1 of the 93-kev y ray in Zn 67 
gives rise to the most precise energy definition 
thus far reported. 2 Despite the numerous diffi- 
culties which beset the experimenter searching 
for the resonance, 3 a small but definite Moss- 
bauer effect has been found. 2 This Letter reports 
measurements on the influence of the nuclear 
Zeeman effect and other perturbing factors upon 
the Mossbauer effect in Zn 87 embedded in an en- 
riched ZnO absorber lattice. 

The relatively high energy of this y ray makes 
it necessary to embed the source and the ab- 
sorber atoms in rigid crystalline lattices, and 
to perform the experiments at low temperatures. 
The first of these requirements was met by using 
ZnO for both the source and the absorber lattices. 
The second requirement was more than satisfied 
by using temperatures below the helium lambda 
transition (2.175°K). In addition to these basic 
requirements, several experimental difficulties 



are consequences of the extreme narrowness of 
the line (4.84 xl0~ u ev). It is accordingly neces- 
sary to take into account the effect of various 
perturbing influences. Here we list the more 
important of these perturbations, and in the fol- 
lowing paragraphs indicate how they enter into 
the design of the experiment. 

One class of shifts arises from the change of 
nuclear mass upon y-ray emission or absorption, 
with a resultant change in the phonon spectrum 
of the lattice. The change in energy of the emit- 
ted y ray is given by 4 



AE = -(E/Mc 2 )<T>, 



(1) 



where E is the y-ray energy, M is the mass of 
the emitting nucleus, and < T) is the expectation 
value for the kinetic energy per atom of the lat- 
tice. If any parameter x should differ between 
source and absorber lattice, the recoil -free 
peaks will occur at different energies in the 
emission spectrum and in the absorption spec- 



561 



213 



Volume 4, Number 11 



PHYSICAL REVIEW LETTERS 



June 1, I960 



tram. The shift is given by 



5(A£) = — (AE)6x. 



(2) 



This expression predicts that a difference in 
Debye temperature between source and absorber 
of only about 1.3°K would cause a shift of one 
linewidth. Thus a change in the average iso- 
topic mass number of 2% would, through the 
mechanism of the Debye temperature (taking 
= 300°K), result in a shift of about four linewidths. 
Similarly, differences in chemical constitution 5 
or lattice defects may be expected to produce 
significant shifts. We shall refer to a shift due 
to difference in isotopic mass between source 
and absorber as the isotopic mass effect. 

In order for the emitted gamma radiation to 
remain unshifted by the recoil of the emitting 
nuclei, it is essential that the recoil momentum 
be absorbed by entire crystallites. For all pre- 
viously observed resonances the minimum size 
of the crystallites was quite small. In the pre- 
sent situation this is no longer true. The recoil 
momentum must be taken up by at least 2 x 10 9 
nuclei, so that the ZnO crystallites must be 
larger than 0.4 micron. Since commercial ZnO 
would normally possess average grain sizes 
smaller than one micron, 8 care must be exer- 
cised to ensure a grain size much larger than 
the above nominal requirement. A sintering pro- 
cess assured that this requirement was met for 
the source. A sample of the enriched absorber 
was studied under an oil immersion microscope. 
No grains smaller than 0.5 micron were ob- 
served, and the majority of the grains were in 
the range of 1 to 2 microns. 

Mechanical vibrations of only 10 " 5 cm/sec 
would produce a Doppler broadening of about one 
linewidth. Since the vibration level in our build- 
ing was nearly 5x10"* cm/sec, the helium cryo- 
stat was shock- mounted and the pumping lines 
carefully decoupled. The source and the ab- 
sorber were clamped rigidly in a single package, 
which was suspended by threads in the helium 
bath. The helium bath was pumped below the 
lambda transition and the nitrogen radiation 
shield was frozen to prevent vibrations from 
boiling liquids and also to place source and ab- 
sorber in an isothermal bath. 

In place of the velocity drive usually used in 
this type of experiment, the resonance was 
shifted by means of the nuclear Zeeman effect. 
The magnetic moment of the Zn 67 nucleus inter- 
acting with an applied magnetic field splits the 



ground state into six components and the excited 
state into four components. Selection rules per- 
mit twelve component gamma transitions, as is 
shown in Fig. 1(a). Here/ =5/2, /g = 3/2, and 
the ground-state gyromagnetic ratio #=0.35. 7 
For the excited state no g value has been meas- 
ured; we adopt arbitrarily the value -1. (The 
negative sign is predicted by the shell model.) 
The source is shielded from the magnetic field 
and hence the emission spectrum is not split. 
For simplicity, the effect of quadrupole interac- 
tions has been omitted. Assuming a shift bE be- 
tween source and absorber, resonance can occur 
for six values of magnetic field. Each component 
has a width which is compounded of the natural 
linewidth and the widths due to residual mechan- 
ical vibrations, quadrupole broadening, etc. 
Even if this width is the same for all components, 
the present method of studying the resonances as 
a function of an external magnetic field will 
cause the apparent or magnetic width of each re- 
sonance peak to be proportional to the field re - 
quired to establish the resonance. Thus reso- 
nances occurring at low values of the applied 
field will appear narrower than those established 
at high fields, and the observed spectrum will 
appear distorted. 
The magnetic field was produced by a small 



(a) 



SE 




--'———-= = 




\»o> 


z 
o 

a. 
ac 
o 
<r> 
a> 




1 







(b) 



FIG. 1. (a) Zeeman splittings of the gamma ray vs 
an applied magnetic field. The gyromagnetic ratio of 
the ground state is 0. 35, and a value of -1.0 has been 
arbitrarily chosen for the excited state, (b) Resonance 
pattern expected from the splittings in (a) in the pre- 
sence of an energy shift 6E between source and ab- 
sorber. In actuality, line broadening would be ex- 
pected to merge closely spaced lines into single peaks. 



562 



214 



Volume 4, Number 11 



PHYSICAL REVIEW LETTERS 



June 1, I960 



solenoid surrounding the absorber. Fields of up 
to 700 gauss could be applied to the absorber, 
while the source was entirely shielded (at oper- 
ating temperatures) by superconducting lead 
foil. Stray fields at the absorber were measured 
to be less than 2 gauss. Fields from the sole- 
noid were prevented from reaching the photo - 
multiplier by a superconducting ring placed in 
the cryostat bottom through which the gamma 
rays emerged. Changes in counting rate due to 
changing fields at the photo multiplier were found, 
even without the superconducting shield, to be 
less than ±0.02%. 

The absorber used in all experiments was 
1.231 g of enriched ZnO (92.4% Zn 87 enrichment) 
pressed with 0.036 g of polyethylene glycol bind- 
er into a button 1.11 cm in diameter. Sources 
were prepared in the Los Alamos cyclotron by 
the reaction Zn^d.njGa 87 on normal sintered 
ZnO. Other reaction products were accounted 
for by a background correction. After bombard- 
ment, the sources were annealed for about one 
hour in air at about 1000°C in order to anneal 
radiation damage and to assure that the Ga 67 
atoms were correctly placed in the ZnO lattice. 
Such a procedure is essential, and lower tem- 
peratures or shorter times yielded erratic re- 
sults, or no resonance whatever. 

Measurements were made of the transmission 
of the absorber vs the applied magnetic field. 
Because of the extremely small change in count- 
ing rate and the large number of counts required 
(typically 10 8 per point), automation of the count- 
ing system and extreme system stability were 
imperative. The 93-kev gamma rays passed 
through thin windows in the cryostat bottom, and 
were detected by a Nal(Tl) scintillation crystal. 
The pulses were amplified and analyzed by a 
single -channel analyzer followed by a scaler 
equipped with a digital recorder. In order to 
eliminate the effect of source decay, all meas- 
urements were taken relative to the counting 
rate when one applied a magnetic field sufficient 
to destroy the resonance entirely. The magnetic 
field sequencing and the scaler were controlled 
by an automatic programmer. Timing was con- 
trolled by a thermostatted quartz crystal fre- 
quency standard with a stated accuracy of 3 parts 
in 10 7 per week. The programming was such as 
to make the measurements insensitive to linear 
drifts. 

Instrument checks were performed by cycling 
the magnetic field (a) at room temperature, (b) 
at low temperature using a nonresonant gamma 



ray, and (c) at low temperature using as an ab- 
sorber a zinc -containing material (gahnite) in 
which no resonance exists. 2 In each case the 
change in counting rate upon application of the 
magnetic field was found to be zero to within 
±0.02%. 

Figure 2 shows the resonance curve obtained. 
The points include corrections for background 
and unresolved nonresonant gamma rays of 
typically 30%. The central features of the data 
are the remarkably large total area under the 
resonance, and the structure and total breadth 
present. The total area is sufficiently large that, 
were there no line broadening, one would esti- 
mate the resonance absorption to be several per- 
cent. Such a value implies a Debye temperature 
for ZnO of about 300°K, which is consistent with 
the results obtained from specific heats. The 
major structural feature of the curve lies in the 
fact that the maximum resonance does not occur 
at zero field, but is shifted to about 10 gauss. 
This result indicates the presence of shifts such 
as the isotopic mass effect mentioned above. 
However, the displacement of the maximum (to 
10 gauss) is smaller than one would predict from 
this effect alone unless an unreasonably large 
value is assumed for the excited state g value. A 
plausible explanation is that chemical shifts 5 are 
superimposed upon the isotopic effect. The pre- 
sence of such shifts is implied by the small re- 
sonance found at zero field using an unenriched 
absorber. 2 Since in tbat case no isotope effects 
were present, only chemical shifts or quadrupole 
broadening can explain the small resonance ob- 



T 


¥ 


1 r/ Mn __, l_ 



SO 100 150 200 250 300 350 400 

FIELD AT ABSORBER (GAUSS) 

FIG. 2. Experimental resonance absorption pattern 
for the 93-kev line of Zn 6T vs the magnetic field applied 
to an enriched ZnO absorber. The source is in zero 
magnetic field. T/p n is the natural level width (in 
ergs) divided by a nuclear magneton (in ergs per 
gauss). 



563 



215 



Volume 4, Number 11 



PHYSICAL REVIEW LETTERS 



June 1, I960 



served. In the present experiments two peaks 
are found separated by only about 40 gauss so 
that it seems unlikely that the broadening is ex- 
treme. The tail of the resonance extends to 
about 400 gauss with no additional structure. 
This may be related to the characteristic of the 
magnetic method that even unbroadened lines 
possess apparent magnetic widths which are pro- 
portional to the applied magnetic field. 

Although the interpretation is admittedly in- 
complete, the extreme sharpness of the reso- 
nance is apparent. In further study, involving 
the development of a Doppler shift drive, we 
hope to measure a number of the energy shifts 
and level splittings mentioned in previous para- 
graphs. 

We wish to thank S, D. Stoddard and R. E. 
Cowan for preparation of the ZnO source buttons 
and for compacting the enriched ZnO absorber. 
The generous cooperation of the cyclotron group 
is gratefully acknowledged. W. E. Keller and 



J. G. Dash each contributed a number of ideas to 
the experiment. 



'Work done under the auspices of the U. S. Atomic 
Energy Commission. 

l R. L. Mossbauer, Z. Physik 151, 124(1958); 
Naturwissenschaften 45, 538(1958); Z. Naturforsch. 
14a, 211 (1959). 

~~*D. E. Nagle, P. P. Craig, and W. E. Keller, Na- 
ture (to be published) . 

3 R. V. Pound and G. A. Rebka, Phys. Rev. Letters 
4, 397 (1960). 

4 R. V. Pound and G. A. Rebka, Phys. Rev. Letters 
4, 337 (1960); B. D. Josephson, Phys. Rev. Letters 
4, 341 (1960). 

s O. C. Kistner and A. W. Sunyar, Phys. Rev. 
Letters 4, 412 (1960). 

8 G. Heiland, E. Mollwo, and F. Stockmann, Solid - 
State Physics , edited by F. Seitz and D. Turnbull 
(Academic Press, New York, 1959), Vol. 8, p. 191. 

T H. Kopfermann, Kernmomente (Akademische Ver- 
lagsgesellschaft, Frankfurt am Main, 1956). 



564 



216 



Mdssbauer Effect in Ferrocyanide 

S. L. Ruby, L. M. Epstein, and K. H. Sun 

Radiation and Nucleonics Laboratory of the Materials Laboratories, 

Westinghouse Electric Corporation, East Pittsburgh, Pennsylvania 

(Received Match 18, 1960) 



IN order to utilize the Mossbauer effect more conven- 
iently, it would be desirable to have either a source or 
an absorber which is monoenergetic. To date, the detailed 
work 1,2 using Fe 87 has employed soft iron sources and ab- 
sorbers ; here, the inner magnetic field creates a hyperfine 
splitting which is disadvantageous for some work. 

During a discussion with the group at the Argonne 
National Laboratories, ferrocyanide was pointed out as a 
possible material with no hyperfine splitting. This follows 
from the fact that potassium ferrocyanide has no magnetic 
moment. 3 Roughly speaking, the electrons from the cyano- 
gen groups completely fill the d-shell of the iron atom and 
eliminate the magnetic behavior. 

The apparatus employs a loud speaker to vibrate the 
Co" source at 25 cps, and a coil in an auxiliary magnetic 
field rigidly attached to the source is used to measure its 
instantaneous velocity. The 14.4-kev y ray is detected con- 
ventionally using a thin Nal(Tl) crystal and a single 
channel analyzer. Its output is used to initiate a multi- 
channel analyzer which has been modified to accept the 
output of the velocity transducer. Thus the complete 
velocity spectrum is scanned every 20 msec, while each 
y ray detected is stored in the appropriate velocity channel. 
The source was made by plating about 300 i*c of Co 57 onto 
soft iron and annealing as suggested by Pound. 4 
Na<Fe(CN) 6 -10H 2 O at 80°K was used as an absorber 
during the measurements reported here. The thickness of 
the absorber is about 100 mg/cm 2 which corresponds to 
0.25 mg/cm 2 of Fe 57 . 

The result of the experiment is given in Fig. 1. For com- 
parison purposes, a spectrum obtained in this apparatus 
using a soft iron absorber at room temperature is also 
shown. Also plotted in'the figure are bars showing the posi- 
tion and relative size of the absorption peaks to be expected 



utilizing the results of Hanna et al., 1 if there is no inner 
field in ferrocyanide. Our present knowledge of instru- 
mental line shape is incomplete, and consequently,- the 
problem of assigning an upper bound to the inner magnetic 
field in ferrocyanide will not be considered here. However, 
if not zero, it is less than 10% of the field in soft iron. 

The foregoing results indicate that in a ferrocyanide 
source the emission from excited Fe 67 would be mono- 
energetic. It would be convenient to have such a source, 
and particularly so if it were effective at room temperature. 
Accordingly, a run using the ferrocyanide absorber was 
made at room temperature. The same peaks are observed 
but with their amplitudes reduced to 60 (±10)% of their 
value at 80°K. An effect of this magnitude can be calcu- 
lated from the formula for the resonant fraction given by 
Pound, 5 



/=exp- 



3 £, 



2 2Mc 2 kd\ 



1 3\ e / J I 



with a Debye 6 of 340 (±40) °K. This suggests that a 
ferrocyanide source will emit about 40% of the 14.4-kev 
y rays without energy change as compared to 60% for 
soft iron. 

However, there is the question of whether the excited 
Fe 67 (produced from Co 57 present as a substitutional im- 
purity) would be present in the proper chemical state. One 
possibility is that cobalt in cyanide complexes tends to go 
to the trivalent oxidation state, whereas the divalent state 
is required for transmutation to ferrocyanide. In addition, 
there is evidence 6 that the cobalto-cyanide complex con- 
tains only five cyanide groups instead of six. 

We would like to thank the members of the Argonne 



217 



LETTERS TO THE EDITOR 






1 ■ 


• i 


1 i ' 


1 1 1 1 1 





- #fA 


Wv } 


A* 


Fe at 300°K vs Na 4 Fe(CN) 6 - 10 H 2 at 80°K 


2 




V 


u 


I 1 


4 




i 1 


V 

1 1 


i i i i i 



4 6 8 10 12 

VELOCITY, MILLIMETERS/ SECOND 



16 



Fig. 1. Percent change in transmission of a Co" (Fe 67 *) source vs velocity of an iron absorber at room temperature and a 
sodium ferrocyanide absorber at 80°K. 



group for their helpful advice, and also Dr. Sergio De 
Benedetti who first suggested the use of a multichannel 
analyzer as a velocity measuring device. 

1 S. S. Hanna et al., Phys. Rev. Letters, 4, 177 (1960). 
1 G. DePasquali et al., Phys. Rev. Letters 4, 71 (1960). 



8 L. Pauling, Nature of the Chemical Bond (Cornell University Press, 
Ithaca, New York, 1960), third ed., p. 166. 

* R. V. Pound and G. A. Rebka, Jr., Phys. Rev. Letters 3, 554 
(1959). 

5 R. V. Pound and G. A. Rebka, Jr., Phys. Rev. Letters 3, 440 
(1959). 

• A. W. Adamson, J. Am. Chem. Soc. 73, 5710 (1951). 



218 

NUCLEAR INSTRUMENTS AND M E TH O D S 12 (1961) 131-137; NORTH-HOLLAND PUBLISHING CO. 



TRANSMISSION AND LINE BROADENING OF RESONANCE RADIATION INCIDENT 
ON A RESONANCE ABSORBER t 

S. MARGULIES and J. R. EHRMAN 

University of Illinois, Urbana, Illinois 

Received 13 February 1961 

The transmission of resonance radiation emitted from a source range of values considered, the transmitted line shape closely 

of finite thickness and passing through an external resonance approximates a Breit-Wigner curve whose width is greater than 

absorber is discussed. The transmission integral is examined for the natural width of the transition. This line broadening, 

both a linear and a Gaussian distribution of radioactive atoms caused by resonance absorption in the source and in the external 

in the source. Several special cases axe presented, and the absorber, is presented graphically as a function of source and 

general case is evaluated by numerical integration. Over the absorber thickness. 

1. Introduction 

R. L. Mossbauer's discovery that recoilless emission and absorption of nuclear gamma radiation can 
occur 1 ) has stimulated a host of recent investigations 2 ). Interest in this process has grown rapidly because 
the Mossbauer effect allows the direct observation of many phenomena formerly thought unmeasurable. 
A terrestial measurement of the gravitational red-shift 3 ), a test of the equivalence principle for rotating 
systems 4 ), and the observation of the Zeeman splitting of excited nuclear levels 5 ), are but a few of the 
experiments made possible by the Mossbauer effect. In addition, the effect is finding many applications in 
the measurement of the internal fields in solids. 

When recoilless emission and absorption of gamma radiation occurs, the conditions for nuclear resonan- 
ce fluorescence are inherently satisfied. The numerous applications of the Mossbauer effect follow from 
the fact that, in such cases, the very narrow lines resulting from transitions from metastable nuclear levels 
can actually be observed: resonance lines with widths in the range 10 -10 to 10 -5 eV, which correspond 
respectively to gamma transitions from levels whose half -lives vary between 10 -5 and 10~ 10 sec, have been 
measured. 

Although experiments involving the Mossbauer effect can be performed with either a transmission or a 
scattering geometry, most of the work done so far has employed the former approach 6 ). A typical trans- 
mission experiment consists of measuring the resonance radiation passing through a resonance absorber, 
as a function of the relative velocity between source and absorber. In this manner, a resonance line shape 
is traced out. Since the great utility of the Mossbauer effect depends upon the measurement of such lines, 
a consideration of the line shape is of interest. In what follows, we will be concerned with the transmission 

■f This work has been supported in part by the joint program 4 ) H. J. Hay, J'. P. Schiffer, T. E. Cranshaw and P. A. Egel- 

of the U.S. Office of Naval Research and the U.S. Atomic staff, Phys. Rev. Letters 4 (1960) 165. 

Energy Commission. s ) R. V. Pound and G. A. Rebka, Jr., Phys. Rev. Letters 3 

(1959) 554; 

x ) R. L. Mossbauer, Z. Physik 151 (1958) 124. G. DePasquali, H. Frauenfelder, S. Margulies and R. N. 

8 ) For a listing of recent experiments, see the bibliography Peacock, Phys. Rev. Letters 4 (1960) 71 ; 

contained in Proceedings of the Allerton Park Conference on S. S. Hanna, J. Heberle, C. Littlejohn, G. J. Perlow, R. S. 

Mdssbauer Effect, University of Illinois, Urbana, Illinois, June Preston and D. H. Vincent, Phys. Rev. Letters 4 (1960) 

5-7, 1960 (unpublished). 177. 

3 ) R. V. Pound and G. A. Rebka, Jr., Phys. Rev. Letters 4 *) H. Frauenfelder, D. R. F. Cochran, D. E. Nagle and R. D. 

(1960) 337. Taylor, Nuovo Cim. 19 (1961) 183. 

131 



219 



132 



S. MARGULIES AND J. R. EHRMAN 



of resonant gamma radiation emitted from a source of finite thicknessf, and passing through a finite 
resonance absorber. In particular, we will consider two types of sources; one in which the radioactive 
atoms are distributed uniformly and one which has a Gaussian distribution of emitting nuclei. 



RESONANCE 
ABSORBER 



l«-jr-JdJ«- 



2. General Formulation 

In our calculation of the transmitted intensity, 
we will assume that a fraction / of all decays 
occur without recoil energy loss 7 ). The resonance 
radiation resulting from these decays will be taken 
to have an emission and absorption spectrum of 
Breit-Wigner shape. The remaining fraction of the 
radiation is non-resonant, and is subject only to or- 
dinary electronic absorption. We will consider a 
source having arbitrary area, and extending in 
depth from x = to x — oo. The distribution of 
emitting atoms along the #-axis will be denoted by 
p(x). We will deal only with the radiation emitted normal to the area of the source, as shown in fig. 1. 
The distribution of the absorbing atoms in both the source and absorber will be taken to be uniform. 

Under these circumstances, the transmission through a resonance absorber of thickness t A , moving 
with a velocity v relative to the source, is given by 




Fig. 1 . Geometry used to calculate the transmission of y-radia- 

tion through a resonance absorber moving with a velocity v 

relative to the source. 



P(y) = e 



(1 



/) J" 'dx p{x)e" sX + / 



x/; 



— f d£ exp [ — /'a«a«a<7o*a 
dxp(x) 



T2/4 



(E — E Q + Sf)* + T2/4 



exp 



(E— £ ) 2 +r a /4J 
T2/4 



(E—Eo + f) 2 + T2/4 



+ (*s 



M) 



(i) 



In this equation, r is the full width at half-height of both the emission and absorption lines which are 
centered about E , and oq is the absorption cross-section at resonancett. The subscripts S and A identify 
the following source and absorber quantities: 

/' — probability of resonance absorption without recoil, 

n = number of atoms per cubic centimeter of volume, 

a = fractional abundance of the atoms which can absorb resonantly, 

fi = ordinary mass attenuation coefficient, evaluated at Eq. 
The quantity^ = {vjc)Eq characterizes the Doppler shift between the source and absorber. 

The first term in (1) represents the transmission of the non-resonant fraction of the radiation, and is 
independent of the Doppler shift Sf. In the second term, which is the resonant contribution, the x-integral 
represents the emission and self-absorption in the source. We will neglect the fi% appearing in the ex- 
ponential of this integral, since the mass absorption is usually much smaller than the resonance absorp- 
tion. The remaining factors in the second term represent the absorption in the external resonance ab- 
sorber. The lower limit on the energy integral has been taken as — oo instead of zero for convenience. 

tf The absorption cross-section at resonance is given by 



7 ) W. Marshall and J. P. Schiffer, The Debye-Waller Factor in 
the Mossbauer Effect, A.E.R.E., Harwell (1960), (unpublished). 

t The case of a beam of y-radiation, having a Breit-Wigner 
energy spectrum, passing through a resonance absorber, has 
been considered by W. M. Visscher in The Evaluation of the 
Transmission Integral, Los Alamos Scientific Laboratory (1959) 
(unpublished) . This corresponds, in effect, to the non-resonant 
absorbing source described in Section 5. 



2»* a 



21* + 1 



2/ + 1 1 + a 



where % is the wavelength of the y-ray, 7* and / are the nuclear 
spins of the initial and final states, respectively, and a is the 
conversion coefficient for the transition. 



220 



TRANSMISSION AND LINE BROADENING 133 

Equation (1) can be seen to be independent of the sign of theDoppler shifts. Since only relative motion 
between source and absorber is pertinent, £f can be included in the absorber part of the transmission 
integral instead of the source part, if desired. 

Equation (1) can easily be generalized for the case when the emission and absorption lines consist of 
more than one component (as when electric or magnetic splitting exists) by forming appropriate sums. 
In this paper we will limit ourselves to the overlap of a single emission line with a single absorption line, 
each centered about Eq. For convenience, we will translate the energy axis so that both lines are centered 
about E == in the absence of any Doppler shift ; that is, (E — Eq) will be replaced by E. 

Sources for Mossbauer experiments are generally prepared in either of two wayst : 

1. The activity is electroplated or otherwise deposited on a source backing. The activity is then diffused 
into the host lattice by heating 8 ). 

2. The activity and the host atoms are co-plated on a backing, thereby building up a lattice containing 
the radioactive atoms as integral parts 9 ). 

The second method produces a source in which the emitting atoms are uniformly distributed in depth. 
The same type of distribution results from the first method if the backing is very thin and the diffusion 
time is very long. On the other hand, if either the backing is thick or the diffusion time is short, then the 
first method produces an activity distribution which is approximately Gaussian. Both types of distribu- 
tion will be discussed below. 

3. Uniform Source Distribution 

We first consider a source of thickness fe, having N radioactive atoms per unit length : 

i N atoms/cm, / s ^ x 2: 
10 x > t 3 . 

Since Nt s is the total number of radioactive atoms and e - ' 1 *'* represents the non-resonant electronic 
absorption in the external absorber, we will deal with a normalized transmission ~P{Sf) defined by 

P(^)=£(^)/(e~" A ' A M s ). (3) 

For the distribution given in (2) the x part of the transmission integral can easily be evaluated, and 
we find 



(4) 



Here, T s — /'s^s^s^s and T A =/'aWa«a^a are effective source and absorber thicknesses, respective- 
ly. The first term in this equation is the non-resonant transmission, and will henceforth be denoted by 
(1 —f)P (uniform). As /isf s approaches zero, this quantity approaches (1 — f). The second term in (4), 

non-res. 

containing an integral over energy, represents the resonant contribution. Before discussing the general 
evaluation of the integral, we will first consider two special cases. 

3.1. THIN SOURCE AND ABSORBER 

When both source and absorber are thin in the sense that T s « 1, T A « 1, we can expand both ex- 
ponentials in the second term of (4) and keep only lowest -order terms in effective thickness. Integrating 
over energy, we obtain 

t Occasionally the target foil, which has been irradiated in a 8 ) R. V. Pound and G. A. Rebka, Jr., Phys. Rev. Letters 3 

pile or in an accelerator beam to produce the desired radio- (1959) 554. 

isotope, may serve directly as a source. In such a case, the 9 ) S. S. Hanna, J. Heberle, C. Littlejohn, G. J. Perlow, 

distribution of emitting atoms can be quite complex, and will R. S. Preston and D. H. Vincent, Phys. Rev. Letters 4 

not be treated here. (1960) 28. 



134 



S. MARGULIES AND J. R. EHRMAN 



P(r) = f (1 -/)P(uniform) + / (l -~) ] 
I non-res. \ 4 / J 



T A 



(yin 2 



221 



(5) 



The bracketed term in this equation represents the transmission when y , the relative Doppler shift 
between source and absorber, is large. This asymptotic value is less than unity because of the self-absorp- 
tion in the source. The second term represents the dip in the transmission due to resonance absorption in 
the external absorber. It is seen that the transmitted line, in the case of thin source and absorber, has a 
Breit-Wigner shape, but has an apparent width r a which is twice that of either the emission or absorption 
spectrum. This broadening results from the overlap of the emission and absorption lines. 

3.2. THIN SOURCE - TRANSMISSION AT ZERO DOPPLER SHIFT 

When the effective source thickness Ts approaches zero (as, for example, in a source which has little 
resonance absorption because as -> 0), the transmission at zero Doppler shift can be found in terms of Ta, 
the effective absorber thickness. Expanding the term containing Ts in (4) and keeping only the lowest- 
order term leads to 

P(0) « (1 -/)P(uniform) + / e"^* Jo(iiT A ) T s -* , (6) 

non-res. 

where Jo is the Bessel function of zero order. Because of the nature of resonance absorption, the resonant 
contribution to the transmission does not decrease exponentially with absorber thickness, but instead, 
shows a saturation behavior. 



3.3. UNIFORM SOURCE - GENERAL CASE 

Attempts to evaluate (4) analytically for arbitrary source and absorber thicknesses have been un- 
successful. Consequently, we have performed a numerical integration on the University of Illinois digital 
computer ILLIAC for values of Ts and 7" a between zero and ten. It has been found empirically that over 
this range the transmitted line is, to a very good degree of approximation, a Breit-Wigner curve whose full 
width at half-height fa, depends upon the value of Ts and Ta. The general variation of transmission with 
Doppler shift & is shown in fig. 2. The results of our numerical integration, in the form of the variation of 
fa/fas a function of Ta with Ts as parameter, are shown in fig. 3. Note that as Ts and Ta both approach 
zero, /"a/f approaches the value two, since the conditions described in section 3.1 are applicable. 

4. Gaussian Source Distribution 

For the case of a Gaussian distribution of radioactive atoms, we will use 

p(x) = (2N/Vn) e'*'"*' atoms/cm, x > . (7) 

Here ts represents a characteristic diffusion depth whose value depends upon the details of the source 
preparation. The above distribution is normalized so that Nts once again represents the total number of 
radioactive atoms. 

Substitution of this Gaussian distribution into (1 ) leads to the following expression, which is normalized 
in the sense defined by (3) : 

r r°° d£ / - r A p2/4 \ 



P(r) = (1 -/) e*** [ 1 -<*Ws/2) ] + t^j 



(E + srf + P2/4 



exp 



( — 

\£2 + 



P2/4/ 



r / T3T2/4 n / T s n/4 \* 

L \2[(E + ^) 2 + P2/4],/ J exp \2[(E + #>)* + n/4]J 



(8) 



where represents the error function, 



*(y) = (2/v^) fV'dw 



222 



TRANSMISSION AND LINE BROADENING 



IN SOURCE 




— NON- 

~ RESONANT | RADIATION LOST 

a " — ~k I BY SELF 

§ RESONANT ' ™°*™» 



— RELATIVE DOPPLER SHIFT ? '-j-f, + 

Fig. 2. Normalized transmission of y-radiation through a 
resonance absorber as a function of the relative Doppler shift 
between source and absorber. The apparent full width at half- 
height of the transmitted line is denoted by .T a . 



6.0 
5.0 
4.0 








f.£ 








r^ 


s^jZ* — 








<^< 


P 


l 


$ 


P 


^ 






/^\ 




SOURCE DISTRIBUTION: 

\N ATOMS /CM 

""'jo 

1 1 1 


1 



Fig. 3. Broadening of the transmitted line for a source having a 
uniform distribution of emitting atoms. 



EFFECTIVE ABSORBER THICKNESS 7". 



Fig. 3. 



The first term in (8) corresponds to the transmission of the non-resonant fraction of the radiation, and 
will be denoted by (1 —f)P (Gaussian). Again, as fi^s approaches zero, the non-resonant contribution 

approaches (1 — /). The resonant contribution to the transmission is contained in the second term of (8). 
Before discussing the general evaluation of this equation, we will consider two special cases. 

4.1. THIN SOURCE AND ABSORBER 

If both source and absorber effective thicknesses satisfy the conditions T$ « 1 , T A « 1 , we can expand 
the exponentials and the error function in the second term of (8), and keep only first-order terms in 
thickness. After integrating over energy, one obtains 

■l±-)]- f I± ! . 

2vVJ 21+ (5»/r)« 



P(s>) 



(1 — /)P(Gaussian) +/(1 



(9) 



In this equation the bracketed term represents the asymptotic transmission as &*-> oo, and differs from 
unity because of self-absorption in the source. The second term results from resonance absorption in the 
external absorber. The transmitted line shape has the Breit-Wigner form, but is twice as wide as either 
the emission or the absorption line. Note that (9) differs from (5) only in that the Gaussian source results 
in more self-absorption than the uniform source. This conclusion follows from the choice of normalization 
for the Gaussian distribution and is subject to the thin source approximation used to derive both equations. 

4.2. THIN SOURCE - TRANSMISSION AT ZERO DOPPLER SHIFT 

Expansion of the appropriate terms to lowest order when Ts -> allows (8) to be evaluated at the point 
!? = 0. Since both the Gaussian distribution of (7) and the uniform distribution described by (2) are 
normalized in the same way, it is not surprising that (6) once again follows for this special case, but with 
P (uniform) replaced by P (Gaussian). 



223 



136 



S. MARGULIES AND J. R. EHRMAN 



4.3. GAUSSIAN SOURCE - GENERAL CASE 

Equation (8), representing the transmission from 
a source which has a Gaussian distribution of radio- 
active atoms, couJd not be evaluated analytically 
for arbitrary values of T$ and Ta- Again, we have 
performed a numerical integration on ILLIAC for 
the range 10 ^ (Ts, Ta) ^ 0. As in the uniform case, 
it has been found empirically that the resulting 
transmitted lines differ but little from Breit-Wigner 
curves whose full widths at half-height vary with 
source and absorber thicknesses (see fig. 2). The 
calculated broadening of the transmitted line as a 
function of Ts and T A is shown in fig. 4. As 7 s and 
Ta approach zero, the conditions of section 4.1 
apply, and T^T approaches the value two. 

5. Non-Resonant Absorbing Sources 

As has been shown, the width of the transmission 

curve obtained from a resonantly absorbing source 

of finite thickness is always greater than twice the 

natural width of the transition. The way to obtain 

the narrowest lines with any given radioisotope is 

to use a source backing which has zero abundance 

of atoms that can absorb resonantly. In this way, 

only the external resonance absorber contributes 

to the line broadening. When this is the case, the ordinary electronic absorption, neglected so far for 

the resonant fraction of the radiation, must be considered. The transmission under these circumstances 

is given by 

W= e- M(1 -4> W e- + ^jr ^(-jgg^xj; ,. yir„,; -) . 00) 















^ 
















fl 


s\/\/ 








# 


//' 


SOURCE 


DISTRIBUTE 

1 


: 

cm ; reo 
1 



EFFECTIVE ABSORBER THICKNESS 



''a"*'*^'* 



Fig. 4. 



Broadening of the transmitted line for a source having 
a Gaussian distribution of emitting atoms. 



-£„) 2 + r*/4/ Jo (E-E ) + !?f + T2/4 

The ^-integral is the same in both the non-resonant and the resonant terms of this equation, and can be 
evaluated for the uniform distribution of (2) and the Gaussian distribution of (7). The transmission, 
normalized according to (3), is found to be 



P{s?) = [{\-i) +//(y)] — 

L fists 
= [(1 — /) + //(*) ]P(uniform) 

non-res. 



uniform source 



and 



where 



P(- 



[ (1 - /) + fl(y) ] [ e i " S ' s) ' (1 - 0(u s t S /2)) ] 
[(1 -/) + fl(sr) ] P(Gaussian) 



Gaussian source 



HD 



2nJ- 



d£ 



(E + sef + T2/4 



exp 



' - T A n/i \ 

>£2 + r 2 / 4 j 



(11) 



(12) 



(13) 



In either case, the shape of the transmitted line is the same, only the amplitude being affected by the 
electronic absorption. This behavior follows from the fact that the ordinary mass attenuation coefficients 



224 



TRANSMISSION AND LINE BROADENING 



137 



are energy independent over the width of the emission and absorption lines. The shape of the transmitted 
line is determined by the energy integral of (13). Since this integral corresponds to the Ts — case for 
either a uniform or a Gaussian source it has, in effect, already been considered. In the results presented 
below, it should be remembered that I{£f) must be substituted into either (1 1 ) or (12) to obtain the trans- 
mission, P(ST). 
When the resonance absorber satisfies the condition T&« 1, the integral of (13) reduces to 



l(y) 



2 i + (^//y 



(14) 



again representing a Breit-Wigner curve whose width at half-height is 2T. 

For arbitrary values of Ta, the transmission also approximates a Breit-Wigner curve, but the apparent 
width r a varies with the absorber thickness in the manner described by the Ts = curve in either fig. 3 
or fig. 4. 

When there exists no Doppler shift between source and absorber, the energy integral of (13) can be 
evaluated : 



1(0) 



Jo(iT A /2) 



(15) 



In section 3.2 and 4.2, where very thin sources were considered, the evaluation of the energy integral 
led to the approximation given in (6). In the case of a non-resonantly absorbing source, no approximations 
are needed, and the result of (15) is exact. Since I{oo) = 1, we can combine (15) with either (11) or (12) 
to get the useful result 



P(oo)-P(0) 
P(oo) 



/[I 



•jr A 



Jo(«V2)] • 



(16) 



Note added in proof: Equation (16), firxt applied to the analysis of 
Mossbauer experiments by the Los Alamos group, is often being 
usedwhenitisnotapplicable.lt must be remembered that (16) is 

1) exact only for non-resonantly absorbing sources; it is a 
good approximation for sources where T$« 1 ; 

2) derived on the assumption that the emission and absorp- 
tion lines overlap exactly at zero relative velocity between 
source and absorber. If, as is often the case, there exists an 
energy shift between emission and absorption spectra, P(0) 
must be replaced by P(Sf ), where Sf is the Doppler shift re- 
quired to produce coincidence; 



3) valid for source and absorber half-widths equal to P, the 
natural width of the transition. It is still correct if these widths 
are not the natural width, provided that they are equal: 
Ps = -Ta.- I q this case, however, the maximum absorption 
cross-section a must be multiplied by the factor P/Pg to keep 
the total absorption constant. 

We wish to emphasize that (16) can be used to extract / and 
/' from absorption measurements only if the conditions assumed 
in the derivation are at least approximately satisfied. 



Acknowledgement 

We wish to thank Professor H. Frauenfelder as well as Dr. E. 
Hetherington for many interesting discussions. 



Liischer, Dr. P. Debrunner, and Mr. J. H. 



225 



Volume 4, Number 4 



PHYSICAL REVIEW LETTERS 



February 15, I960 



POLARIZED SPECTRA AND HYPERFINE STRUCTURE IN Fe 57 * 

S. S. Hanna, J. Heberle, C. Littlejohn, G. J. Perlow, R. S. Preston, and D. H. Vincent 

Argonne National Laboratory, Lemont, Illinois 

(Received January 26, 1960) 



The observation in this laboratory 1 of the 
polarization of the resonance radiation 2 " 5 emitted 
by the 14-kev level of Fe 57 has led to a study of 
polarization in the hyperfine spectrum of the 
resonant absorption. The apparatus and the 
method of producing and detecting polarization 
were the same as used in reference 1 except that 
the source and the absorber were mounted on 
separate Alnico magnets. The magnet carrying 



the absorber was attached firmly to the bed of 
the lathe used in our previous work. 4 The other 
magnet holding the source was fastened securely 
to the carriage of the lathe. The detector of 
radiation (40 -mil Nal) was mounted on the axis 
determined by the source and absorber and it 
was well shielded from magnetic fields. 

The motion of the carriage provided uniform 
velocities of the source, and the polarized spec- 



177 



226 



Volume 4, Number 4 



PHYSICAL REVIEW LETTERS 



February 15, I960 



tra were obtained by measuring the transmission, 
with crossed or parallel magnetizations in source 
and absorber, as a function of the velocity of the 
source. The operation of the lathe was made 
automatic so that the carriage (source) moved 
to and fro at a predetermined speed. During the 
"to" motions the pulses from the detector were 
recorded in the lower channels of a 2 56 -channel 
analyzer; and during the "fro" motions they 
were accumulated in the upper channels of the 
analyzer. In a single run, therefore, the trans- 
mission was measured for a positive and for an 
equal negative velocity. 

The spectra obtained in this way are shown in 
Fig. 1. Since no significant differences were 
observed for positive and negative velocities, 
the spectra have been folded about zero velocity. 
The spectrum obtained with source and absorber 




5 10 15 

SPEED IN mm/sec 

FIG. 1. Hyperfine spectra of Fe 5T . Top: unpolarized. 
Middle: magnetization in source and absorber parallel. 
Bottom: magnetization in source and absorber perpen- 
dicular. The ordinate is in units of 2000. 



magnetized perpendicular to each other differs 
markedly from that obtained with parallel mag- 
netizations. For comparison, a spectrum is 
shown for an unmagnetized source and absorber. 
It is seen that the hyperfine spectrum consists 
of six prominent lines instead of the four pre- 
viously reported. 3 ' 5 It is clear therefore that the 
earlier interpretation based on the existence of 
only four lines is incorrect. 

The level diagram of Fe 57 which seems to pro- 
vide a satisfactory explanation of the spectra in 
Fig. 1 is shown on the left in Fig. 2. In the upper 
right are given the hyperfine components for Ml 
radiation. (We have found little need to introduce 
a significant amount of £2.) The intensities of 
the components are those appropriate to a ran- 
dom orientation of the internal magnetic fields at 
the nuclei. At the lower right are shown the 
components for the case in which the internal 
fields have been aligned. The intensities given 
are for radiations emitted perpendicular to the 
aligned field. The direction of polarization of 
each component relative to the direction of the 
aligned field is indicated by the symbol II or ± . 

If one takes a hyperfine pattern and moves it 
over itself, one obtains the hyperfine spectrum, 



-3/2 
1/2 
1/2 
3/2 



• ■ — 

* — t 

* f 





Li 



UN POL. 



II I 

_Li 



6 - 90*. POL. 



1/2 



1/2 



.57 



FIG. 2. Level diagram of Fe" on which the dis- 
cussion is based. Upper right: unpolarized hyperfine 
pattern. The numbers give the relative intensities. 
Lower right: polarized hyperfine pattern {9 = 90°) . 
The symbols II and x stand for polarization parallel 
or perpendicular to the aligned field. 



178 



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Volume 4, Number 4 



PHYSICAL REVIEW LETTERS 



February 15, I960 



each line in the spectrum arising from the coinci- 
dence of hyperfine components in emission and 
absorption. At the top in Fig. 3 is shown the 
predicted spectrum of unpolarized radiation. In 
the middle is given the spectrum for the case in 
which the internal fields in source and absorber 
are aligned parallel to each other. In this case 
a line in the absorption spectrum will appear 
only if the respective hyperfine components have 
the same polarization. If, on the other hand, 
these polarizations are perpendicular, then the 
line will appear in the absorption spectrum only 
if the internal fields in source and absorber are 
aligned at right angles. The spectrum predicted 
for this case is shown at the bottom in Fig. 3. 
The intensities given in Fig. 3 are those nomin- 
ally expected for a thin absorber. In addition 
it is assumed that a line which should appear 
only with one orientation of the fields will actually 
be present to the extent of about 10% with the 
other orientation, because of incomplete align- 
ment of the fields in source and absorber. The 
spectra in Fig. 3 are in good qualitative agree-, 
ment with the observations in Fig. 1. 

The hyperfine pattern of six components pro- 
duces, in all, eight lines in the absorption spec- 
trum. However, the splittings in the ground 
state and in the excited state are such that two 
doublets are formed which are not resolved in 
the unpolarized spectrum. The resolved peaks 
are numbered from one to six in Fig. 3. One 
member of the doublet in line 2 is too weak to 
affect the position of the peak. Thus, the spac- 
ings between line 1 and 2, 4 and 5, and 5 and 6 
should be equal to the splitting of the ground state. 
The spacing between lines 2 and 4 gives the split- 
ting of the ground state. Line 3 is a doublet, one 
member of which should appear in the spectrum 
with parallel fields, the other in the spectrum 
with crossed fields. The separation in the doublet 
is equal to 2^ -g , where g x and g are the split- 
tings of the excited and ground level, respective- 
ly. We have measured this doublet separation 
with some care by observing the shift in line 3 
in going from one polarized spectrum to the 
other. The separation is (0.5 ±0.1) mm/sec. We 
have also measured the separation between lines 
1 and 2 more carefully than shown in Fig. 1 and 
obtained g x =(2.23 ± 0.03) mm/sec. Hence g 
= (3. 96 i 0.10) mm/sec. 

Ludwig and Woodbury 8 have recently obtained 
an accurate determination of the magnetic mo- 
ment of the ground state. If we use their value of 
+(0.0903 ±0.0007) nm, the above measurements 





■ '1 

3 4 5 
2 


5 
UNPOL. 




■1 | 


' 1 


1 1 


1 ' 

1 


1 

1 



9i 



V«o 



*Wo 



FIG. 3. Spectra predicted by the scheme in Fig. 2. 
Top: unpolarized. Middle: magnetizations parallel. 
Bottom: magnetizations perpendicular. The main 
peaks are numbered from one to six. The symbols 
g and #i represent the gyromagnetic ratios of ground 
and excited levels, respectively. 



give -(0.153 ±0.004) nm for the magnetic moment 
of the excited state, and a value of (3.33 ±0.10)xl0 5 
oersteds for the effective magnetic field at the iron 
nucleus. We note the opposite sign of the mag- 
netic moment, which is an important feature of 
the above interpretation. 

We would like to thank B. F. Martinka and H. W. 
Ostrander for generous assistance in the mechan- 
ical aspects of our work. We are grateful to 



179 



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Volume 4, Number 4 



PHYSICAL REVIEW LETTERS 



February 15, 1960 



G. W. Ludwig and H. H. Woodbury for communi- 
cating their results to us. 



Work performed under the auspices of the U. S. 
Atomic Energy Commission. 

*G. J. Perlow, S. S. Hanna, M. Hamermesh, 
C. Littlejohn, D. H. Vincent, R. S. Preston, and 
J. Heberle, Phys. Rev. Letters 4, 74 (1960). 

2 J. P. Schiffer and W. Marshall, Phys. Rev. Let- 



ters 3, 556 (1959). 

S R. V. Pound and G. A. Rebka, Jr., Phys. Rev. 
Letters 3, 554 (1959). 

4 S. S. Hanna, J. Heberle, C. Littlejohn, G. J. Per- 
low, R. S. Preston, and D. H. Vincent, Phys. Rev. 
Letters4, 28 (I960). 

5 G. DePasquali, H. Frauenf elder , S. Margulies, and 
R. N. Peacock, Phys. Rev. Letters 4, 71 (1960). 

6 G. W. Ludwig and H. H. Woodbury, Phys. Rev. 
(to be published). 



180 



229 



Volume 4, Number 8 



PHYSICAL REVIEW LETTERS 



April 15, I960 



EVIDENCE FOR QUADRUPOLE INTERACTION OF Fe 57w , 
AND INFLUENCE OF CHEMICAL BINDING ON NUCLEAR GAMMA -RAY ENERGY" 

O. C. Kistner and A. W. Sunyar 

Brookhaven National Laboratory, Upton, New York 

(Received March 30, 1960) 



The recoil -free emission and resonant absorp- 
tion 1 of the 14.4 -kev nuclear gamma ray of Fe 57 , 
has been used to determine the quadrupole coup- 
ling for the 3/2- excited state of Fe 57 bound in 
Fe 2 3 , and to measure an energy shift of this 
nuclear gamma ray which is attributed to effects 
of chemical binding. This effect is corollary to 
the effects of chemical environment on internal 
conversion coefficients 2 and on electron capture 
disintegration rates. 3 These measurements also 
yield the value of the internal magnetic field at 
the position of the Fe 57 nucleus when it is bound 
in antiferromagnetic Fe 2 3 . 

The Ml emission line of Fe 57 bound in ordinary 
metallic iron is split into six components by the 
magnetic hyperfine interaction. 4 The resonant 
absorption of this emission spectrum by Fe 57 
bound in Fe 2 3 has been examined, as well as 
the much simpler absorption pattern which re- 
sults when the "unsplit" emission line from Fe 57 
bound in a stainless steel lattice 5 is used. Be- 
cause the interpretation of the two sets of meas- 
urements agree, only the latter measurements 
are presented in this Letter. The former meas- 
urements, in which a Co 57 source co -plated with 
iron onto 1-mil copper was used, will only 
briefly be remarked upon. 

The ~2-mg/cm 2 Fe 2 3 absorber used in these 
measurements contained Fe 57 enriched to -30%. 
The source consisted of Co 57 plated onto 0.001- 
inch stainless steel (25% Cr, 20% Ni). After 
plating, this source was annealed for one hour 
at 900°C in a hydrogen atmosphere. All meas- 
urements were made with source and absorber 
at a temperature of 25°C. 

In order to obtain a Doppler shift of the emis- 
sion line, a uniform motion was obtained by 
coupling a pneumatically driven cylinder to 
another cylinder filled with oil, the ports of 
which were connected via a needle valve. A 
wide range of uniform velocities could be selec- 
ted by adjusting this valve. The direction of 
source travel was reversed automatically by 
means of microswitches. Additional micro- 
switches, set to exclude the region of nonuniform 
motion near the travel limits, were used to pro- 
vide gate signals for the counters. The distance 
of travel between the limits of the counting gates 
was 0.973 cm. Source velocity was determined 

412 



by counting the cycles from a 1000 cps tuning 
fork oscillator during the time between the gate 
limits. 

The 14.4 -kev gamma ray was detected with a 
Nal(Tl) scintillation counter. The phosphor was 
2 mm thick and 1.5 inches in diameter. The 
counter face was located 5.4 cm above the upper 
limit of vertical travel of the source. The ab- 
sorber was placed 4.7 cm from the counter face. 
A single -channel pulse -height analyzer selected 
the 14.4 -kev gamma ray photopeak. The outputs 
of this analyzer and the 1000 cps clock were 
switched between two pairs of scalers so as to 
record counting rate and velocity separately for 
both directions of source motion. 

Figure 1 shows the counting rate (in arbitrary 
units) as a function of source velocity relative to 
the Fe 2 3 absorber for the stainless steel source. 
Absorption of the "unsplit" emission line at each 
of the six possible absorption energies of Fe 57 in 
Fe 2 3 is evident. The lack of symmetry of the 
absorption pattern about zero velocity shows im- 
mediately that one is not dealing simply with a 
magnetic hyperfine splitting pattern. 

The velocities at which absorption peaks occur 
are given in Table I. Experimental values have 
been corrected by -2.5% to take account of the 
effect of geometry on our velocity scale. These 
absorption line velocities may be fitted precisely 
in terms of an energy level diagram as shown 
in Fig. 2. It is necessary to introduce an energy 
shift AE = A£ x + AE 2 between the center of gravity 
of the absorption lines of Fe 57 in Fe 2 3 and the 
emission line of Fe 57 in stainless steel. In addi- 
tion, an energy shift e, of positive or negative 
sign, on the individual m states of the excited 
level is required. This is interpreted as being 
caused by a quadrupole interaction when Fe 57 is 
bound in Fe 2 3 . A least -squares fit to our data 
yields the following splitting parameters (in 
"velocity units") for the two Fe 57 nuclear states 



g ' = 0.611 ±0.005 cm/sec, 

&' = 0.345 ±0.003 cm/sec, 

AE= 0.047 ±0.003 cm/sec, 

e=0.012 ±0.003 cm/sec. 



230 

Volume 4, Number 8 PHYSICAL REVIEW LETTERS April 15, I960 



150 


















w 145 


















O 

> 140 


















i 


















135 




• 












" 




-10 


-8 


-6 


-4-2024 
VELOCITY OF SOURCE (mm /sec) 


6 


8 


10 



ENERGY CHANGE , E 



FIG. 1. The absorption by Fe ' bound in Fe 2 3 of the 14.4-kev gamma ray emitted in the 
decay of Fe^ m bound in stainless steel as a function of relative source- absorber velocity. 
Positive velocity indicates a motion of source toward absorber. 



These data indicate that the ratio g ' /g t ' is 
1.77 ± 0.02, in excellent agreement with the re- 
sults of Hanna et al., 4 and that the internal mag- 
netic field in antiferromagnetic Fe 2 3 is larger 
than in ferromagnetic iron by the factor 1.547 
± 0.022. If we take 3.33 x 10 5 oersteds as the 
value of the internal field 4 in Fe, we find a value 
of 5.15 x10 s oersteds at the Fe nucleus in Fe 2 3 . 
It is worth noting that the absorption peaks in 
Fig. 1 (reading from left to right) should have 
"thin absorber" intensity ratios of 3:2:1:1:2:3. 
However, the relative intensities found for the 
absorption peaks clearly exhibit evidence of sa- 
turation due to the rather large effective thick- 
ness of the enriched Fe 2 3 absorber. Neverthe- 
less, the experimental intensity ratios are such 
that an inverted hyperfine pattern is required for 



Table I. Velocities of maximum absorption (cor- 
rected for geometry) for a stainless steel source and 
Fe 2 3 absorber. The negative signs indicate motions 
of source away from absorber. 



Absorption line 



Relative source-absorber 
velocity (cm/sec) 



1/2+ — 3/2+ 
1/2+ — 1/2+ 
1/2+ — 1/2- 
1/2-— 1/2+ 
1/2- — 1/2- 
1/2- — 3/2- 



-0.789 
-0.417 
-0.076 
+0.192 
+0.536 
+0.859 



the 3/2- state, providing confirmation for this 
fact as previously reported. 4 
Let us now consider the effect of quadrupole 




k 

r 




K 






- f 


{ 


















«c 


• 








\ 





Fe METAL 



STAINLESS STEEL 



FIG . 2 . Schematic representation of the ground and 
14.4-kev excited states of Fe 57 bound in ordinary iron, 
Fe 2 3 , and stainless steel. This diagram illustrates 
the details of magnetic hyperfine splitting, quadrupole 
interaction, and energy shifts due to chemical binding 
effects . 



413 



231 



Volume 4, Number 8 



PHYSICAL REVIEW LETTERS 



April 15, I960 



interaction. For a pure quadrupole spectrum, 
in the case of axially symmetric field gradients, 
the shift e for the substates of a spin / state is 
given by 8 



e 2 gQ 
4/(27-1) 



[3m 2 -/(/ + !)], 



where q denotes (l/e)(d 2 V/dz 2 ). All other sym- 
bols have the conventional meanings. When 
Zeeman splitting is combined with quadrupole 
interaction, the precise energy shifts for the 
individual m states due to quadrupole interaction 
depend upon the orientation of the magnetic axis 
relative to the axis of symmetry for the electric 
field gradient, and are in general not the same 
for all m states. 7 However, for a case such as 
we deal with here, where the quadrupole inter- 
action is small compared to the magnetic inter- 
action, the absolute magnitude of e is the same 
for all m states of the spin 3/2 state within the 
accuracy of our measurements, independent of 
the orientation of the crystal axes. Thus, we 
find € = \\e 2 qQ\ =0.012±0.003 cm/sec (5.75xl0~ 9 
ev). Since our measurements indicate that the 
shift of magnitude e is negative for the rn =±3/2 
substates and positive for the m =±1/2 substates, 
the product e 2 qQ is negative. The absolute value 
is suggestive of a small quadrupole moment for 
p e 57w although this statement cannot be ampli- 
fied further without knowledge of electric field 
gradients in Fe 2 3 . 

In our earlier absorption measurements on 
Fe 2 3 using the emission lines of Fe 57 bound in 
ordinary iron, we find an energy shift upward 
(see Fig. 2) of AE^O.04 cm/sec. Within the 
accuracy of our measurements, the splitting 
parameters g ', g^, and e remain the same. A 
separate measurement of the absorption by Fe 57 
bound in stainless steel of the emission lines of 
Fe 57 in ordinary iron confirms the implied ex- 
istence of an energy shift downward (see Fig. 2) 
of AE 2 = 0.01 cm/sec in this case. 

The existence of an energy shift AE =(2.26±0.15) 
xlO" 8 ev between stainless steel and Fe 2 3 has 
been definitely established. We will now discuss 
the ways in which differences in chemical environ- 
ment may produce such a shift. (1) Source and 
absorber will in general have different Debye 
temperatures. Since the nucleus in its excited 
state has a slightly greater rest mass than when 
it is in its ground state, the energy difference 
between the nuclear states will decrease from its 
value for an "unbound" system by different 



amounts because of the difference in the zero- 
point energies. This reduction in energy is lar- 
ger for the substance having the higher Debye 
temperature. (2) In addition, when a lattice is 
at a finite temperature, the energy difference 
between the nuclear states is reduced further by 
virtue of the previously reported temperature 
effect. 8 Since the magnitude of this reduction 
depends upon the integral of the specific heat of 
the lattice up to the temperature of the substance, 
it will be larger for the substance having the 
lower Debye temperature. We expect this tem- 
perature effect to be small, although it should be 
noted that it acts to reduce the energy shift be- 
tween different substances which results from 
zero-point energy differences. The sign and 
magnitude of the observed shift between stain- 
less steel and Fe 2 3 would require that Fe 2 3 
have a considerably lower Debye temperature 
than stainless steel or ordinary iron. Were this 
the case, the Debye -Waller factor would materi- 
ally depress the recoil -free resonant yield in 
Fe 2 3 , a result which is not indicated by the data. 
(3) When chemical environment is altered, a 
nuclear isotope shift may result. This effect has 
its origin in the change in the electronic wave 
functions over the region of space occupied by 
the nucleus, s electrons may be expected to con- 
tribute most to this effect. Since s electrons are 
in effect removed in going from Fe in metal to 
Fe in Fe 2 3 , a smaller charge radius for Fe 57W 
than for Fe 57 in its ground state would produce a 
shift in the observed direction. The direction of 
the observed energy shift AE requires the pre- 
sence of the nuclear isotope shift, since the zero- 
point energy shift is in the opposite direction. 
This nuclear isotope shift is similar to the iso- 
meric isotope shift 9 observed in the optical spec- 
trum of Hg 197 . 

We wish to emphasize two additional points 
about recoil -free emission and absorption ex- 
periments when source and absorber are chemi- 
cally different. First, the existence of energy 
shifts introduces asymmetries into the absorp- 
tion pattern which make it essential not to com- 
bine data taken at equal velocities of opposite 
sense. Second, any recoil -free absorption at 
zero velocity is accidental. In general, there- 
fore, a search for this effect by comparing ab- 
sorption at zero velocity with absorption at a 
large relative velocity between source and ab- 
sorber will not yield significant results unless 
source and absorber are identical chemically. It 



414 



232 

Volume 4, Number 8 PHYSICAL REVIEW LETTERS April 15, I960 



is amusing to note that in the absorption by Fe 2 3 
of the emission lines of Fe 57 in ordinary iron, 
two such accidental coincidences do indeed occur 
to give substantial absorption at zero velocity. 

We wish to thank many of our colleagues, 
particularly M. Goldhaber and J. Weneser, for 
interesting discussions, and G. K. Wertheim 
for providing us with a sample of the particular 
stainless steel used in his measurements. 5 



Work done under the auspices of the U.S. Atomic 
Energy Commission. 

*R. L. Mbssbauer, Z. Physik 151, 124 (1958). 

2 K. T. Bainbridge, M. Goldhaber, and E. Wilson, 
Phys. Rev. 90, 430 (1953). 



3 R. F. Leininger, E. Segre, and C. Wiegand, Phys. 
Rev. 76, 897 (1949). 

4 S. S. Hanna, J. Heberle, C. Littlejohn, G. J. Per- 
low, R. S. Preston, and D. H. Vincent, Phys. Rev. 
Letters 4, 177 (1960). 

5 G. K. Wertheim (private communication) has in- 
formed us that Fe 57 in stainless steel exhibits a single 
absorption line. [See G. K. Wertheim, this issue, 
Phys. Rev. Letters 4, 403(1960).] 

6 T. P. Das and E. L. Hahn, Nuclear Quadrupole 
Resonance Spectroscopy , Solid State Physics, Suppl. 1 
(Academic Press Inc., New York, 1958). 

7 P. M. Parker, J. Chem. Phys. 24, 1096 (1956). 

8 R. V. Pound and G. A. Rebka, Jr. , Phys. Rev. 
Letters 4, 274 (1960). 

9 A. C. Melissinos andS. P. Davis, Phys. Rev. 115 , 
130(1959). 






415 



233 



Volume 6, Number 2 



PHYSICAL REVIEW LETTERS 



January 15, 1961 



ELECTRIC QUADRUPOLE SPLITTING AND THE NUCLEAR 
VOLUME EFFECT IN THE IONS OF Fe 87 t 

S. DeBenedetti, G. Lang, and R. Ingalls* 
Carnegie Institute of Technology, Pittsburgh, Pennsylvania 
(Received December 19, 1960) 



A study of the Mo'ssbauer spectra of Fe 57 in 
chemical compounds has revealed some striking 
regularities. Typical transmission curves for 
Fe ++ and Fe +++ ionic compounds relative to an 
unsplit room -temperature stainless steel source 
are shown in Figs. 1 and 2, respectively. Both 
patterns exhibit a shift of the center of gravity 
from the zero of velocity, and the curve of Fe ++ 
shows the two absorption lines characteristic of 
a pure nuclear electric quadrupole interaction. 
The quantities 6 and A£, which characterize 
these effects, are defined in the figures; their 
values for various compounds are shown in 
Table I. It is noteworthy that the value of 6 and 
the low -temperature value of &E are, for a given 
iron ion, relatively independent of the chemical 
combination. 

The center of gravity displacement, or chemi- 
cal shift, 1 arises from two mechanisms: (a) the 
second order Doppler shift, 2 ' 3 which is caused by 
lattice vibrations, and is a function A(T s ,T a ,9 s ,6 a ) 
of the source and absorber temperatures and 
Debye temperatures, and (b) the nuclear volume 
effect, 4 ' 5 which represents the change, from 
source to absorber, in the modification of the 
nuclear transition energy caused by electrons 
overlapping the finite nucleus. The total shift in 
a Mo'ssbauer spectrum may be written: 

6 *A(T s , T q , e s , e a ) + B(R e 2p -R g 2p )[* a 2 (0) - */(())], 

(1) 




-2 



-I +1 + 2 +3 +4 

SOURCE VELOCITY (mm/sec) 



FIG. 1 . Mbssbauer spectrum of the Fe ++ ion in an 
absorber of FeSo 4 '7H 2 at nitrogen temperature using 
a room-temperature stainless steel source. The pat- 
tern exhibits the chemical shift 6 and the electric quad- 
rupole splitting A£ of the excited state of the Fe 5T nu- 
cleus. The velocity is positive for the source approach- 
ing the absorber . 



where R e and R g are the charge radii of the nu- 
clear excited and ground states. *^(0) is the 
s -electron density at the position of the nucleus. 
B is a constant which is 1.76 xlO 9 cm/sec for 
iron when R^p an( j ^2( ) are expressed in atomic 
units, p =(1 - oPZ 2 ) m is a relativistic factor 5 equal 
to 0.982 for iron. 

An estimate of Rg/R e may be made if one neg- 
lects the influence of the crystalline surroundings 
and uses wave functions 6 for the free iron ions. 
Using (1) and the room -temperature shifts (which 
are less sensitive to differences in 6 than are the 
low-temperature shifts), we obtain 



6 «B(R 

+++ e 



«/ P)[ * ++ ~ (0) 



We have neglected the small difference A ++ -A +++ 
Watson's wave functions yield *++ 2 (0)< * +++ 2 (0); 
the extra electron in Fe ++ apparently shields the 
3s electrons slightly. Equation (2) yields R g /R e 
= 1.001. 

Watson 6 ' 7 has also calculated wave functions 
for the outer electron configurations 3d 8 4s 2 and 
3d 8 for iron. Using an equation similar to (2) one 
obtains the shifts expected for each case: 

6 =-0.178 cm/sec for 3d 4 4s 2 , 



Fe 



+0.182 cm/sec for 3d 8 . 



When metallic iron is used as an absorber with 
the stainless steel source at room temperature, 
a shift of +0.01 cm/sec results. Since this value 
is midway between the above calculated values, 



uji.ao 




\ r 


■ . ■ ' 


M.IO 

O 
O 




V 




UJlOO 




«*T 




H 


-2 -1 ( 


\ +| +2 


+3 



£ SOURCE VELOCITY (mm/feec) 

FIG. 2. Mbssbauer spectrum of the Fe +++ ion in a 
FeCl s absorber at nitrogen temperature using a room- 
temperature stainless steel source . The velocity is 
positive for the source approaching the absorber . 



60 



234 



Volume 6, Number 2 



PHYSICAL REVIEW LETTERS 



January 15, 1961 



Table I. A list of quadrupole splittings A£ and chemical shifts 6 relative to the emission line from a stainless 
steel source at room temperature. 



Temperature 



A£(cm/sec) 



6(cm/sec) 



Fe 2 3 




Room 


0.024 ±0.003 


0.047 ±0.003 


a 


Fe 2 3 




Room 


0.024 


0.050 


b 


Fe 2 (S0 4 ) 3 




Room 





0.064 


b 


Fe 2 (S0 4 ) 3 




Room 





0.055 ±0.005 


this work 


Fe 2 (S0 4 ) 3 




Nitrogen 





0.065 ±0.005 


this work 


FeCl 3 




Room 





0.045 ±0.005 


this work 


FeCl 3 




Nitrogen 





0.065 ±0.005 


this work 


FeCl 2 -4H 2 




Room 


0.300 ±0.005 


0.135 ±0.005 


this work 


FeCl 2 -4H 2 




Nitrogen 


0.310 ±0.005 


0.145 ±0.005 


this work 


FeS0 4 -7H 2 




Room 


0.320 ±0.005 


0.140 ±0.005 


this work 


FeSCy7H 2 




Nitrogen 


0.360 ±0.005 


0.150 ±0.005 


this work 


Fe(NH 4 ) 2 (S0 4 ) 2 


•6H 2 


Room 


0.175 ±0.005 


0.140 ±0.005 


this work 


Fe(NH 4 ) 2 (S0 4 ) 2 


•6H 2 


Nitrogen 


0.270 ±0.005 


0.150 ±0.005 


this work 


FeF, 




Room 


0.268 




c 



a See reference 4. 
I. Solomon, Proceedings of the Allerton Park Conference on Mbssbauer Effect, University of Illinois, 



Urbana, Illinois, June 5-7, 1960 (unpublished), Sec. lie. 
"G. K. Wertheim, Phys. Rev. 121, 63 (1961). 



it strongly suggests that metallic iron has an 
outer electron configuration equivalent to 3d 7 4s 1 . 

Although the electric quadrupole splitting seems 
to be temperature dependent, it is roughly the 
same at low temperatures for all the Fe ++ com- 
pounds listed in Table I. This suggests that the 
relevant field gradient is mainly caused by the 
ion alone. The free Fe ++ ion has an outer elec- 
tron configuration of 3d 6 and is in a 5 Z? state. 
Five of these 3d electrons have their spins aligned 
and together form a spherically symmetric dis- 
tribution of charge. The sixth electron has oppo- 
site spin and also a choice of the five possible 
values of mj. The crystal symmetry dictates the 
m/ combination the ion will actually take. For 
sufficiently low symmetry the lowest energy state 
will be a linear combination of the two states with 
mi equal to 2 and -2. Assuming this, and neg- 
lecting all other possible effects of the crystalline 
environment, it is possible to calculate the ionic 
field gradient at the nucleus to within the factor 



(1 -y), where y is the Sternheimer antishielding 
factor. Taking A£ to be 0.3 cm/sec one obtains 
I Q(l -y) I s= 0.1 b, where Q is the nuclear electric 
quadrupole moment for the excited state of Fe". 

The temperature dependence of the quadrupole 
splitting could be caused by thermal excitation of 
the Fe ++ ion into a state which yields a lower field 
gradient at the nucleus. For sufficiently rapid 
ionic transitions, the nucleus will respond to the 
time average of the field gradient. This interpre- 
tation indicates that the level splitting of the Fe ++ 
ion in Fe(NH 4 ) 2 (S0 4 ) 2 -6H 2 is sufficiently small 
that significant excitation occurs at room tempera- 
ture. 

It is noted that there is very little quadrupole 
splitting in the case of Fe +++ . Presumably there 
is negligible field gradient from this ion, since 
its 3d 5 electron configuration gives a 6 S state. 
The splitting that does occur 4 is therefore caused 
by field gradients from the neighboring ions rather 
than from the Fe +++ ion itself. 



61 



235 

Volume 6, Number 2 PHYSICAL REVIEW LETTERS January 15, 1961 

We are grateful to Professor J. Michael Rad- Sec. He. 
cliff e for some very helpful discussions. 2 R. V. Pound and G. A. Rebka, Jr. , Phys. Rev. 
Letters 4, 274 (1960). 

'''This work was supported by the Office of Naval 3 B. D. Josephson, Phys. Rev. Letters 4, 341 (1960) . 

Research. 4 0. C. Kistner and A. W. Sunyar, Phys. Rev. 

'Submitted by R. Ingalls in partial fulfillment of the Letters 4, 412 (1960). 
requirements for the degree of Doctor of Philosophy at 5 G. Breit, Revs. Modern Phys. 30, 507 (1958). 

the Carnegie Institute of Technology. 6 R. E. Watson, Technical Report No. 12, Solid- 

*0. C Kistner, Proceedings of the Allerton Park State and Molecular Theory Group, Massachusetts 

Conference on Mossbauer Effect, University of Illi- Institute of Technology (unpublished), 
nois, Urbana, Illinois, June 5-7, 1960 (unpublished), 7 R. E. Watson, Phys. Rev. 119, 1934 (1960). 



62 



236 



Volume 6, Number 3 



PHYSICAL REVIEW LETTERS 



February 1, 1961 



INTERPRETATION OF THE Fe 57 ISOMER SHIFT 

L. R. Walker, G. K. Wertheim, and V. Jaccarino 

Bell Telephone Laboratories, Murray Hill, New Jersey 

(Received January 3, 1961) 



We have made a systematic study of the Mo'ss- 
bauer effect 1 of Fe 57 in di- and tri-valent iron 
compounds and in d-group metals. The observed 
"isomer shift" 2 measures the total s -electron 
density at the nucleus. The shift in compounds 
is shown to depend mainly upon the 3d configura- 
tion of iron involved and to a lesser extent upon 
the "chemical" bond. The Hartree-Fock calcula- 
tions of Watson 3 on the various 3d configurations 
of iron are combined with the data on the shifts 
in the most ionic compounds to obtain a calibra- 
tion of the shift in terms of s -electron density. 
This enables us to estimate the difference in 
charge radius of the ground state and isomeric 
state of Fe 57 . We associate the shift of an Fe 
solute ion in a d -group metal with the addition of 
some fraction of a 4s electron to an Fe 3d 7 con- 
figuration; an estimate of the 4s-electron wave 
function density at the nucleus from the Fermi- 
Segre-Goudsmit (FSG) formula 4 enables this 
fraction to be determined. 

Kistner and Sunyar 5 first observed in the recoil - 
free emission and resonant absorption of the 
14.4-kev nuclear gamma ray of Fe S7 , that the 



energies of the emitted and absorbed gamma rays 
were noticeably different if the emitter and ab- 
sorber were two dissimilar lattices containing 
iron. The origin of this effect is as follows. 

The ground and isomeric levels of the nucleus 
have different effective charge radii; the elec- 
trostatic interaction with the electronic charge 
is then different in the two states and the gamma - 
ray energy is consequently changed (relative to 
its value for a point nucleus) by an amount pro- 
portional to the s -electron density at the nucleus. 
If the s-electron density is different for the ab- 
sorber and emitter, the difference in gamma - 
ray energies, E a -E e , is defined as the isomer 
shift. 

E -E =f7rZe 2 [K io 2 -K 2 ][l^(0) l 2 -l^(0) I 2 ], (1) 



is 



gr 



and ground states and l^(0) a l 2 and \ip(0) e I 2 are 
the total s-electron densities at the nucleus for 
absorber and emitter, respectively. 9 It is to be 
noted that the sign of the shift has its origin in 
the fact that the level energy is lower the more 



237 



Volume 6, Number 3 



PHYSICAL REVIEW LETTERS 



February 1, 1961 



compact the charge distribution. 

An examination of the restricted Hartree-Fock 
calculations of Watson 3 shows that there are 
significant differences in the value of l^ 3S (0)l 2 
for different 3d configurations of iron; the change 
in l^ lS (0)l 2 and l^ 2 s(°)l 2 is substantially smaller. 

It is perhaps worthwhile to point out that un- 
restricted Hartree-Fock calculations, which ex- 
hibit the imbalance of up and down spin density 
(exchange polarization effect), appear to give the 
same total density at the nucleus as the restricted 
procedure. 7 The variation of I</j sS (0)I 2 is such as 
to correspond to different degrees of shielding 
of 3s by 3d electrons. To calibrate the observed 
shifts in terms of total s -electron density, we 
associate the difference in the shifts for the most 
ionic Fe 2+ and Fe 3+ compounds with the differ- 
ence in Watson's values of £ n =Jl^ ws (0)l 2 for 3d 6 
and 3d 5 configurations. When an Fe atom is in- 
troduced into a d -group host metal, £ w I <p ns {0) I 2 
will also contain a contribution from 4s conduc- 
tion electrons. Hartree-Fock calculations do 
not exist for single 4s electrons outside 3d n con- 
figurations, but the FSG 4 formula, combined with 
the known term value of 3d w 4s configurations, 8 
provides an entirely adequate estimate of l^ 4S (0) I 2 . 

In Fig. 1 is shown a possible interpretation of 
the observed shifts making use of the above ideas. 
The total s-electron density in atomic units is 
plotted as ordinate. Watson's values for 
2En=?l^ ns (0)l 2 for the Fe 3d 7 * configurations 
from n = 4 to n = 8 are indicated on the left. The 
scale on the right for MOssbauer center- of- gravi- 
ty shifts relative to stainless steel is established 
by identifying the shifts in the most ionic Fe 2+ 
and Fe s+ with Watson's densities for 3d and 3d 5 , 
respectively. The solid straight lines represent 
s-electron densities for hypothetical 3d n \s x con- 
figurations. They are drawn on the assumption 
that the density for such a configuration is of the 
form \ip(3d n )\ 2 +x\ip 4S (0)\ 2 , where l^ 4S (0)l 2 is 
calculated from the FSG formula for a single 4s 
electron outside the 3d" configuration. This 
assumes no screening of inner s electrons by 
the single 4s electron. 9 Curves for the configur- 
ations 3d 8_x 4s Jf and 3d 7 "*4s* are indicated by 
dashed lines and were obtained by extrapolation 
and interpolation of the FSG formula. The ex- 
perimental data are given in Table I. 

The shifts for Fe in various metals have been 
represented as horizontal lines of a length suf- 
ficient to cover what appear to be the most plaus- 
ible configurations. Since for Fe in Fe metal 
there are certainly 8 electrons to be accounted 




FIG. 1. A possible interpretation of the Fe" Mbss- 
bauer isomer shifts In various solids. The total s-elec- 
tron density is plotted as a function of the percentage 
of 45 character for various d -electron configurations. 
The reasons for placing the experimental data on given 
theoretical curves are discussed in the text. The con- 
stant C = 11 873 a t ~ 3 . 



for, the evidence seems to point clearly to the 
configuration 3d 7 4s for this case. The small- 
ness of the spread in shifts between the different 
metals indicates strongly that the configuration 
of the solute Fe is substantially the same in the 
metals investigated. For the Fe 2+ and Fe s+ 
compounds, the experimental data are entered 
upon the 3d 6 4s x and 3d 5 4s* curves, respectively. 
This is consistent with the idea that in these 
covalent compounds the 4s atomic orbitals are 
partially occupied by electrons from the ligand 
ions (bonding orbitals). 10 ' 11 The data for the 
"ferro" and "ferri" cyanides of potassium have 
not been plotted since it is not clear how to fit 
them into the above scheme. Indeed, since they 
have ground states which do not follow Hund's 
rule, it is unlikely that Watson's calculations are 
applicable to them. The fact that the isomer 
shifts in both cyanides are very small relative 
to each other and to stainless steel appears to be 
fortuitous. 

From the observed isomer shift and the cal- 
culated difference in I ^<0) I 2 for the d 5 and d e 
free-ion configurations we may compute the dif- 
ference of the excited and ground-state charge 



99 



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Volume 6, Number 3 



PHYSICAL REVIEW LETTERS 



February 1, 1961 



Table I. Observed shifts in gamma-ray energy in 
various iron compounds and d -group metals measured 
relative to type 310 stainless steel. (Source and ab- 
sorber at room temperature.) The measured shifts 
contain contributions from the second-order Doppler 
shift and the zero-point energy shift in addition to the 
isomer shift discussed here. Measurements at 77°K a 
indicate that the second-order Doppler shift contribu- 
tions are small compared to the total shift. More- 
over, since the Doppler shifts are likely to be of sim- 
ilar magnitudes in the compounds considered, they 
will cancel, since we are ultimately concerned only 
with differences in the shifts between similar mater- 
ials. The same is true of the zero-point energy shift. 
The uncertainty in comparing metals with salts is 
considerably greater. To convert the shifts from 
cm/sec to Mc/sec, one should multiply the values by 
116. The errors indicated reflect the uncertainty in 
the last significant figure. 



Shift 
(cm/sec) 


3d* 


FeF 2 (single crystal) 


0.140 ±5 


3d* 


KFeFj 


0.139 ±5 


3d* 


FeSCv 7H 2 


0.140 ±5 


3d* 


FeCl 2 - 4H 2 


0.130 ±5 


3d* 


FeS 


~0.11 ±1 


3d 6 


Fe 2 (S0 4 ) 3 - eHjO 


0.052 ±5 


3d 5 


Fe 2 0, b 


0.047 ±5 


3d 8 


Yttrium-iron garnet, 
octahedral 


0.057 ±5 


3d' 


Yttrium -iron garnet, 

tetrahedral 


0.026 ±5 




FeS 2 (pyrites) 


0.048 ±5 




FeS 2 (marcasite) 


0.048 ±5 


Metals 


Fe b 


0.015 ±5 




Co 


0.012 ±5 




Ni 


0.015 ±5 




Mn 


-0.008 ±2 




Cr 


-0.005 ±2 




Mo 


-0.001 ±2 


Cyanides 


K4Fe(CN) 6 '3H 2 


0.0083 ±10 




K s Fe(CN) s 


0.0000 ±10 


a R. L. Ingalls, G. Lang, and S. 
Am. Phys. Soc. 5, 429 (1960). 
"See reference 5. 


DeBenedetti, Bull. 



radii using Eq. (1), assuming the usual charge 
radius dependence on mass number, i.e., 
R = \.20A U3 xlO~ i3 cm. We obtain t>R/R = 1.8xl0' 3 
as the fractional change in the charge radius, 
with the effective radius of the ground state 
larger than that of the excited state. This result 
is not unexpected 12 in sign and magnitude. The 
shell model predicts that the ground state for 3 
odd neutrons outside a closed shell (28 neutrons), 
corresponding to a hole in the 2p 3f2 shell, is one 
for which 7 = 3/2, as is the case for the 14-kev 
excited state. The large spin-orbit coupling pre- 
cludes the possibility of a 2p v2 configuration 
being the ground state. 

Using the radial moments for an isotropic 
square well 13 and even assuming a proton ex- 
citation corresponding to A/ = 3 or 4, one still 
obtains a value of only 2xl0" 3 for bR/R. (The 
more likely case of a neutron excitation would 
leave the charge radius unchanged in a first ap- 
proximation.) It is most unlikely therefore that 
the ground state is a simple shell state. It is 
interesting to note that the charge radius change 
expected for the addition of one particle (isotope 
shift at A = 57) is bR/R = $(6A/A) = 5.9xl0" 3 . 

In the isomeric transition of Hg 197 (I =1/2 — 
/ ex = 13/2), it was estimated 2 that the charge re- 
distribution corresponded to an increase of one- 
fourth to one -fifth of that observed experimentally 
for the addition of one neutron at A =157. (The 
excited state has the larger charge distribution 
in this case.) Since both states are identifiable 
as single -particle neutron states, it is clear that 
a general charge redistribution accompanies the 
isomeric transition. 

We would like to thank W. E. Blumberg, A. M. 
Clogston, and M. Goldhaber for several critical 
discussions. 



'See reference 6. 



! R. L. Mossbauer, Z. Physik 151, 124(1958). 

isomer shift was first observed in the optical spectra 
of Hg 197 (ground state /= 1/2, excited state /= 13/2); 
A. C. Melissinos and S. P. Davis, Phys. Rev. 115 , 
130 (1959). If one accepts "isotope shift" to be the 
proper name for charge redistribution effects resulting 
from addition of particles, the name "isomer shift" is 
the logical choice for the effect discussed herein. Un- 
fortunately, the phrase "chemical shift" has been used 
to describe Mossbauer isomer shifts in the past. 

3 R. E. Watson, Solid State and Molecular Theory 
Group, Technical Report No. 12, Massachusetts Insti- 
tute of Technology, June 15, 1959 (unpublished). 

4 E. Fermi and E. Segre, Z. Physik 82, 729 (1933); 
S. A. Goudsmit, Phys. Rev. 43, 636 (1933). 

s O. C. Kistner and A. W. Sunyar, Phys. Rev. Letters 
4, 412(1960). A temperature-dependent shift had been 



100 



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Volume 6. Number 3 



PHYSICAL REVIEW LETTERS 



February 1, 1961 



previously observed and identified as a second-order 
Doppler shift resulting from the thermal motion of the 
atoms. See R. V. Pound and G. A. Rebka, Jr., Phys. 
Rev. Letters 4, 274(1960). 

•i. Solomon, Compt. rend. 250 , 3828 (1960). Apart 
from the numerical constant which was not given in 
this reference, our Eq. (1) is the same as the result 
given there. 

T R. E. Watson and A. J. Freeman, Phys. Rev. 120 , 
1125(1960). 

8 C. E. Moore, Atomic Energy Levels , National 
Bureau of Standards Circular No. 467 (U. S. Govern- 



ment Printing Office, Washington, D. C. 1952), Vol. 2 

9 R. E. Watson, Phys. Rev. 119, 1934 (1960). An 
estimate of the charge |fy,(0) | 2 for the inner s electron 
was made from this reference in which the configura- 
tions 3d n ~ 2 4s l are considered. 

,0 J. S. Van Wieringen, Discussions Faraday Soc. 
No. 19, 118(1955). 

u The effect of covalency on the occupation of d orbit - 
als is neglected. 

12 M. Goldhaber (private communication). 

,3 J. Eisinger and V. Jaccarino, Revs. Modern Phys. 
30, 528 (1958). 



101 



240 

PHYSICAL REVIEW 
LETTERS 



Volume 4 



APRIL 1, I960 



Number 7 



APPARENT WEIGHT OF PHOTONS* 

R. V. Pound and G. A. Rebka, Jr. 

Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 

(Received March 9, 1960) 



As we proposed a few months ago, 1 we have now 
measured the effect, originally hypothesized by 
Einstein, 2 of gravitational potential on the ap- 
parent frequency of electromagnetic radiation by 
using the sharply defined energy of recoil -free 
y rays emitted and absorbed in solids, as dis- 
covered by MCssbauer. 3 We have already re- 
ported 4 a detailed study of the shape and width of 
the line obtained at room temperature for the 
14.4-kev, 0. 1 -microsecond level in Fe 57 . Partic- 
ular attention was paid to finding the conditions 
required to obtain a narrow line. We found that 
the line had a Lorentzian shape with a fractional 
full -width at half -height of 1.13 xlO -12 when the 
source was carefully prepared according to a 
prescription developed from experience. We have 
also investigated the 93-kev, 9.4 -microsecond 
level of Zn 67 at liquid helium and liquid nitrogen 
temperatures using several combinations of 
source and absorber environment, but have not 
observed a usable resonant absorption. That work 
will be reported later. The fractional width and 
intensity of the absorption in Fe 57 seemed suffi- 
cient to measure the gravitational effect in the 
laboratory. 

As a preliminary, we sought possible sources 
of systematic error that would interfere with 
measurements of small changes in frequency using 
this medium. Early in our development of the in- 
strumentation necessary for this experiment, we 
concluded that there were asymmetries in, or 
frequency differences between, the lines of given 
combinations of source and absorber which vary 
from one combination to another. Thus it is ab- 



solutely necessary to measure a change in the 
relative frequency that is produced by the per- 
turbation being studied. Observation of a fre- 
quency difference between a given source and 
absorber cannot be uniquely attributed to this 
perturbation. More recently, we have discovered 
and explained a variation of frequency with tem- 
perature of either the source or absorber. 8 We 
conclude that the temperature difference between 
the source and absorber must be accurately 
known and its effect considered before any mean- 
ing can be extracted from even a change observed 
when the perturbation is altered. 

The basic elements of the apparatus finally 
developed to measure the gravitational shift in 
frequency were a carefully prepared source 
containing 0.4 curie of 270-day Co 57 , and a care- 
fully prepared, rigidly supported, iron film ab- 
sorber. Using the results of our initial experi- 
ment, we requested the Nuclear Science and 
Engineering Corporation to repurify their nickel 
cyclotron target by ion exchange to reduce cobalt 
carrier. Following the bombardment, in a special 
run in the high -energy proton beam of the high- 
current cyclotron at the Oak Ridge National 
Laboratory, they electroplated the separated 
Co 57 onto one side of a 2-in. diameter, 0.005-in. 
thick disk of Armco iron according to our pre- 
scription. After this disk was received, it was 
heated to 900°-1000°C for one hour in a hydrogen 
atmosphere 6 to diffuse the cobalt into the iron 
foil about 3xl0" 5 cm. 

The absorber made by Nuclear Metals Inc., 
was composed of seven separate units. Each 



337 



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Volume 4, Number 7 



PHYSICAL REVIEW LETTERS 



April 1, I960 



unit consisted of about 80 mg of iron, enriched 
in Fe 57 to 31.9%, electroplated onto a polished 
side of a 3 -in. diameter, 0.040-in. thick disk of 
beryllium. The electroplating technique required 
considerable development to produce films with 
absorption lines of width and strength that satis - 
fied our tests. The films finally accepted, reso- 
nantly absorbed about 1/3 the recoil -free y rays 
from our source. Each unit of the absorber was 
mounted over the 0.001 -in. Al window of a 
3 in. x 1/4 in. Nal(Tl) scintillation crystal in- 
tegrally mounted on a Dumont 6363 multiplier 
phototube. The multiplier supply voltages were 
separately adjusted to equalize their conversion 
gains, and their outputs were mixed. 

The required stable vertical baseline was con- 
veniently obtained in the enclosed, isolated tower 
of the Jefferson Physical Laboratory. 7 A statis- 
tical argument suggests that the precision of a 
measurement of the gravitational frequency shift 
should be independent of the height. Instrumental 
instability but more significantly the sources of 
systematic error mentioned above are less criti- 
cal compared to the larger fractional shifts ob- 
tained with an increased height. Our net operating 
baseline of 74 feet required only conveniently 
realizable control over these sources of error. 



The absorption of the 14.4 -kev y ray by air in 
the path was reduced by running a 16-in. diam- 
eter, cylindrical, Mylar bag with thin end win- 
dows and filled with helium through most of the 
distance between source and absorber. To 
sweep out small amounts of air diffusing into the 
bag, the helium was kept flowing through it at a 
rate of about 30 liters/hr. 

The over -all experiment is described by the 
block diagram of Fig. 1. The source was moved 
sinusoidally by either a ferroelectric or a moving- 
coil magnetic transducer. During the quarter of 
the modulation cycle centered about the time of 
maximum velocity the pulses from the scintilla- 
tion spectrometer, adjusted to select the 14.4- 
kev y-ray line, were fed into one scaler -while, 
during the opposite quarter cycle, they were fed 
into another. The difference in counts recorded 
was a measure of the asymmetry in, or frequency- 
shift between, the emission and absorption lines. 
As a precaution the relative phase of the gating 
pulses and the sinusoidal modulation were dis- 
played continuously. The data were found to be 
insensitive to phase changes much larger than 
the drifts of phase observed. 

A completely duplicate system of electronics, 
controlled by the same gating pulses, recorded 



HYDRAULIC MASTER 



RACK S PINION 
CLOCK DRIVE 




^#fr 



m^ 



JULTL^LTL 



MONITOR CHANNEL 




AMPLIFIER 
AND 
PULSE HEIGHTr - 
SELECTOR 



COINCIDENCE 
CIRCUIT 



DISCRIMINATOR 



AMPLIFIER 

AND 

PULSE HEIGHT 

SELECTOR 



MERCURY 
RELAYS 



COINCIDENCE 



__ CIRCUIT _ MERCURY 

RELAYS 



DISCRIMINATOR 



| COUNTER | 



|cqunter| 



FIG. 1. A block diagram of the over-all experimental arrangement. The source and ab- 
sorber-detector units were frequently interchanged. Sometimes a ferroelectric and some- 
times a moving-coil magnetic transducer was used with frequencies ranging from 10 to 50 cps. 



338 



242 

Volume 4, Number 7 PHYSICAL REVIEW LETTERS April 1, I960 



data from a counter having a 1-in. diameter, 
0.015-in. thick Nal(Tl) scintillation crystal cov- 
ered by an absorber similar to the main absorb- 
er. This absorber and crystal unit was mounted 
to see the source from only three feet away. 
This monitor channel measured the stability of 
the over -all modulation system, and, because of 
its higher counting rate, had a smaller statistical 
uncertainty. 

The relation between the counting rate differ- 
ence and relative frequency shifts between the 
emission and absorption lines was measured 
directly by adding a Doppler shift several times 
the size of the gravitational shift to the emission 
line. The necessary constant velocity was intro- 
duced by coupling a hydraulic cylinder of large 
bore carrying the transducer and source to a 
master cylinder of small bore connected to a 
rack -and -pinion driven by a clock. 

Combining data from two periods having Dop- 
pler shifts of equal magnitude, but opposite sign, 
allowed measurement of both sensitivity and 
relative frequency shift. Because no sacrifice 
of valuable data resulted, the sensitivity was 
calibrated about 1/3 of the operating time which 
was as often as convenient without recording the 
data automatically. In this way we were able to 
pUminate errors due to drifts in sensitivity such 
as would be anticipated from gain or discrimina- 
tor drift, changes in background, or changes in 
modulation swing. 

The second order Doppler shift resulting from 
lattice vibrations required that the temperature 
difference between the source and absorber be 
controlled or monitored. A difference of 1°C 
would produce a shift as large as that sought, so 
the potential difference of a thermocouple with 
one junction at the source and the other at the 
main absorber was recorded. An identical sys- 
tem was provided for the monitor channel. The 
recorded temperature data were integrated over 
a counting period, and the average determined 
to 0.03°C. The temperature coefficient of fre- 
quency which we have used to correct the data, 
was calculated from the specific heat of a lattice 
having a Debye temperature of 420°K. Although 
at room temperature this coefficient is but weakly 
dependent on the Debye temperature, residual 
error in the correction for, or control of, the 
temperature difference limits the ability to meas- 
ure frequency shifts and favors the use of a large 
height difference for the gravitational experiment. 

Data typical of those collected are shown in 
Table I. The right -hand column is the data after 



correction for temperature difference. All data 
are expressed as fractional frequency shift xlO 15 . 
The difference of the shift seen with y rays rising 
and that with y rays falling should be the result 
of gravity. The average for the two directions 
of travel should measure an effective shift of 
other origin, and this is about four times the 
difference between the shifts. We confirmed that 
this shift was an inherent property of the particu- 
lar combination of source and absorber by meas- 
uring the shift for each absorber unit in turn, 
with temperature correction, when it was six 
inches from the source. Although this test was 
not exact because only about half the area of each 
absorber was involved, the weighted mean shift 
from this test for the combination of all absorber 
units agreed well with that observed in the main 
experiment. The individual fractional frequency 
shifts found for these, for the monitor absorber, 
as well as for a 11.7-mg/cm 2 Armco iron foil, 
are displayed in Table II. The considerable var- 
iation among them is as striking as the size of 
the weighted mean shift. Such shifts could result 
from differences in a range of about 11 % in ef- 
fective Debye temperature through producing 
differences in net second order Doppler effect. 
Other explanations based on hyperfine structure 
including electric quadrupole interactions are 
also plausible. Although heat treatment might be 
expected to change these shifts for the iron -plated 
beryllium absorbers, experience showed that the 
line width was materially increased by such 
treatment, probably owing to interdiffusion. The 
presence of a significant shift for even the Armco 
foil relative to the source, both of which had re- 
ceived heat treatments, suggests that it is unlikely 
one would have, without test, a shift of this sort 
smaller than the gravitational effect expected in 
even our "two-way" baseline of 148 feet. The 
apparently fortuitous smallness of the shift of 
the monitor absorber relative to our source cor- 
responds to the shift expected for about 30 feet 
of height difference. 

Recently Cranshaw, Schiffer, and Whitehead 8 
claimed to have measured the gravitational shift 
using the y ray of Fe 57 . They state that they be- 
lieve their 43% statistical uncertainty represents 
the major error. Two much larger sources of 
error apparently have not been considered: (1) 
the temperature difference between the source 
and absorber, and (2) the frequency difference 
inherent in a given combination of source and 
absorber. From the above discussion, only 0.6°C 
of temperature difference would produce a shift 



339 



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Volume 4, Number 7 



PHYSICAL REVIEW LETTERS 



April 1, I960 



Table I. Data from the first four days of counting. The data are expressed as fractional frequency difference 
between source and absorber multiplied by 10 15 , as derived from the appropriate sensitivity calibration. The 
negative signs mean that the y ray has a frequency lower than the frequency of maximum absorption at the ab- 
sorber. 







Shift 


Temperature 




Net 




Period 


observed 


correction 




shift 






Source at bottom 










Feb. 


22, 5 p.m. 


-11.5 ±3.0 
-16.4 ±2. 2 a 
-13.8±1.3 
-11.9±2.1 a 
-8.7 ±2.0 a 


-9.2 
-5.9 
-5.3 
-8.0 
-10.5 






-20. 7 ±3.0 
-22.3 ±2.2 
-19.1 ±1.3 
-19.9±2.1 
-19.2 ±2.0 


Feb. 


23, 10 p.m. 


-10.5 ±2.0 

Source at top 


-10.6 


Weighted 


average 1 


-21.0±2.0 
-19.7 ±0.8 


Feb. 


24, a.m. 


-12.0±4.1 
-5.7±1.4 
-7.4±2.1 a 
-6.5±2.1 a 

-13.9 ±3. l a 
-6.6±3.0 


-8.6 
-9.6 
-7.4 
-5.8 
-7.5 
-5.7 






-20.6 ±4.1 
-15.3 ±1.4 
-14. 8 ±2.1 
-12.3±2.1 
-21.4±3.1 
-12.3±3.0 


Feb. 


25, 6 p.m. 


-6.5 ±2.0 a 
-10.0±2.6 


-8.9 
-7.9 


Weighted 


average = 


-15.4±2.0 
-17.9 ±2.6 
-15.5±0.8 










Mean shift = 


-17.6 ±0.6 








Difference of averages = 


-4.2 ±1.1 



a These data were taken simultaneously with a sensitivity calibration. 



Table II. Data on asymmetries of various absorb- 
ers in apparent fractional frequency shift multiplied 
by 10 15 . In the third column we tabulate the Debye 
temperature increase of the absorber above that of the 
source which could account for the shift. 



Absorber 


(AiV^)xio 15 


A8j) in "K 


No. 1 


-8.4*2.5 


+15 ±4 


No. 2 


-24 ±3. 5 


+41 ±6 


No. 3 


-28 ±3. 5 


+48 ±6 


No. 4 


-19 ±3. 5 


+33 ±6 


No. 5 


-24 ±3. 5 


+41 ±6 


No. 6 


-17 ±2.5 


+29 ±4 


No. 7 


-19 ±3. 5 


+33 ±6 


Weighted mean 






of No. 1-No. 7 


-19 ±3.0 


+33 ±5 


Monitor absorber 


+0.55 ±0.15 


-0.95 ±0.26 


Armco foil 


+10±3.5 


-17 ±6 



as large as the whole effect observed. Their 
additional experiment at the shortened height dif- 
ference of three meters does not, without con- 
comitant temperature data, resolve the question 



of inherent frequency difference. Their stated 
disappointment with the over -all line width ob- 
served would seem to add to the probability of 
existence of such a shift. They mention this 
broadening in connection with its possible influ- 
ence on the sensitivity, derived rather than 
measured, owing to a departure from Lorentzian 
shape. Clearly such a departure is even more 
important in allowing asymmetry. 

Our experience shows that no conclusion can 
be drawn from the experiment of Cranshaw et al. 

If the frequency -shift inherent in our source- 
absorber combination is not affected by inversion 
of the relative positions, the difference between 
shifts observed with rising and falling y rays 
measures the effect of gravity. All data collected 
since recognizing the need for temperature cor- 
rection, yield a net fractional shift, -(5.13 ±0.51) 
x1 q-i5 The error assigned is the rms statistical 
deviation including that of independent sensitivity 
calibrations taken as representative of their re- 
spective periods of operation. The shift observed 
agrees with -4.92xl0" ls , the predicted gravita- 
tional shift for this "two-way" height difference. 



340 



244 



Volume 4, Number 7 



PHYSICAL REVIEW LETTERS 



April 1, I960 



Expressed in this unit, the result is 

(Ai/) /(AvL. =+1.05±0.10, 
v 'exp theor 

where the plus sign indicates that the frequency 
increases in falling, as expected. 

These data were collected in about 10 days of 
operation. We expect to continue counting with 
some improvements in sensitivity, and to reduce 
the statistical uncertainty about fourfold. With 
our present experimental arrangement this should 
result in a comparable reduction in error in the 
measurement since we believe we can take ade- 
quate steps to avoid systematic errors on the re- 
sulting scale. A higher baseline or possibly a 
narrower y ray would seem to be required to 
extend the precision by a factor much larger than 
this. 

We wish to express deep appreciation for the 
generosity, encouragement, and assistance with 
details of the experiment accorded us by our 
colleagues and the entire technical staff of these 
laboratories during the three months we have 



been preoccupied with it. 



Supported in part by the joint program of the Office 
of Naval Research and the U. S. Atomic Energy Com- 
mission and by a grant from the Higgins Scientific 
Trust. 

*R. V. Pound and G. A. Rebka, Jr. , Phys. Rev. 
Letters 3, 439 (1959). 

2 A. Einstein, Ann. Physik 35, 898(1911). 

3 R. L. Mossbauer, Z. Physik 151, 124(1958); 
Naturwissenschaften 45, 538 (1958); Z. Naturforsch. 
14a, 211 (1959). 

*R, V. Pound and G. A. Rebka, Jr. , Phys. Rev. 
Letters 3, 554 (1959). 

6 R. V. Pound and G. A. Rebka, Jr., Phys. Rev. 
Letters 4, 274 (1960). 

6 We wish to thank Mr. F. Rosebury of the Research 
Laboratory of Electronics, Massachusetts Institute of 
Technology, for providing his facilities for this treat- 
ment. 

T See E. H. Hall, Phys. Rev. 17, 245 (1903), first 
paragraph. 

8 T. E. Cranshaw, J. P. Schiffer, and A. B. White- 
head, Phys. Rev. Letters 4, 163(1960). 



245 



Volume 4, Number 4 



PHYSICAL REVIEW LETTERS 



February 15, I960 




MEASUREMENT OF THE RED SHIFT IN AN ACCELERATED SYSTEM 
USING THE MOSSBAUER EFFECT IN Fe 57 

H. J. Hay, J. P. Schiffer,* T. E. Cranshaw, and P. A. Egelstaff 

Atomic Energy Research Establishment, Harwell, England 

(Received January 27, 1960) 

In an adjoining paper 1 an experiment is described 
in which the change of frequency in a photon pass- 
ing between two points of different gravitational 
potential has been measured. Einstein's principle 
of equivalence states that a gravitational field is 
locally indistinguishable from an accelerated sys- 
tem. It therefore seemed desirable to measure 
the shift in the energy of 14-kev gamma rays 
from Fe" in an accelerated system. In order to 
do this we have plated a Co 57 source on to the 
surface of a 0.8 -cm diameter iron cylinder. 
This cylinder was rigidly mounted between two 
Dural plates which also held a cylindrical shell 
of Lucite, 13.28 cm in diameter and 0.31 cm 
thick, concentric with the iron cylinder. An iron 
foil 3.5 mg/cm 2 thick and enriched in Fe 57 to 
50% was glued to the inside surface of the Lucite. 
This assembly was mounted in a neutron chopper 
drive unit 2 and rotated at angular velocities up 
to 500 cycles per second. The gamma rays 
passing through the absorber were detected in a 
xenon-filled proportional counter. A schematic 
diagram of the apparatus is shown in Fig. 1 . 

The expected shift can be calculated in two 
ways. One can treat the acceleration as an effec- 
tive gravitational field and calculate the difference 
in potential between the source and absorber, or 
one can obtain the same answer using the time 
dilatation of special relativity. The expected 
fractional shift in the energy of the gamma ray 
is (/e i 2 -/e 2 2 )oj 2 /2c 2 =2.44xl0- 20 u) 2 . 

The number of gamma rays as a function of 
angular velocity is shown in Fig. 2. In a sep- 
arate measurement the counting rate as a func- 
tion of radial velocity was determined for this 
same source and absorber. It was found that 
with the source moving rapidly the counting rate 
was 1.29 times what it was with the source sta- 




FIG. 2. Comparison of the calculated curve with 
experimental points. The statistical errors of each 
point are indicated. The curve was calculated from 
the parameters given in the text. 

tionary. The measured full width of the resonance 
was 0.38 mm/sec. The curve calculated from 
these parameters is also shown in Fig. 2. The 
sensitivity of the equipment to vibrations was 
tested by vibrating the shaft of the rotor with 
frequencies corresponding to the rotational fre- 
quencies involved, and negligible effect was ob- 
served. Changes in counting rate due to forces 
on the absorber were also found to be negligible. 

It appears that the observed effect is in reason- 
able agreement with expectations. The size of 
the shift of the gamma -ray energy in the effec- 
tive gravitational field of a rotating system is 
in agreement with that due to the terrestrial 
gravitational field, within the accuracy of the 
measurements. The present experiment is ex- 
pected to be improved when a more pure source 
is available for reasons stated in the previous 
paper. It will also be necessary to study further 
the factors which could influence the absorption 
process, including changes in the magnetic 
hyperfine fields due to the high velocities. 

We would like to acknowledge helpful and illu- 
minating discussions with Dr. J. S. Bell, Dr. W. 
Marshall, and Dr. T. Skyrme. We would also 
like to thank Dr. E. Bretscher for his support 
and encouragement. 



FIG. 1 . Schematic diagram of the experimental 
equipment. 



John Simon Guggenheim fellow, on leave from 



165 



246 



Volume 4, Number 4 



PHYSICAL REVIEW LETTERS 



February 15, 1960 



Argonne National Laboratory, Lemont, Illinois. 

'T. E. Cranshaw, J. P. Schiffer, and A. B. White- 
head, preceding Letter (Phys. Rev. Letters 4, 163 



(I960)]. 

2 Egelstaff, Hay, Holt, Raffle, and Pickles, J. Inst. 
Elec. Engrs. (London) (to be published). 



166 



247 



Absence of Doppler Shift for Gamma Ray Source 
and Detector on Same Circular Orbit 

By D. C, CHAMPENEY and P. B. MOON 

Department of Physics, University of Birmingham 
MS. received 29th August 1960 

Abstract. An experiment is described showing that for a source and absorber of 
57 Fe placed at opposite ends of a rotor the Mossbauer absorption is unaffected by 
rotation. This is contrary to the situation with the source at the centre when 
relativistic effects cause a frequency shift between source and absorber. 



Reporting a test of the effect of circular motion on the resonant frequency of 
the gamma-ray transition in 57 Fe, Hay, Schiffer, Cranshaw and Egelstaff 
^ (1960) point out that one can treat the acceleration as an effective gravita- 
tional field and calculate the frequency shift from the difference of potential 
between source and absorber, or one can obtain the same answer by using the time 
dilatation of special relativity. 

For their arrangement, with the source at the centre and the absorber at the 
periphery of the rotating system, the same result also follows from the argument 
that since source and absorber have relative velocity v (<^c) in a direction perpen- 
dicular to the line joining them, there exists a transverse Doppler effect giving a 
fractional frequency shift v 2 /2c 2 . 

It is perhaps surprising that the naive use of this formula, without any account 
being taken of acceleration, should give the correct answer ; an indication of the 
subtleties that may be involved is obtained by considering source and absorber 
to move on the same circle, e.g. at opposite points on the periphery. Their 
pseudo-gravitational potentials are equal, so are their time-dilatations, yet their 
relative velocity is 2v. 

We are indebted to several of our colleagues for interesting comments on this 
problem, involving such matters as Coriolis forces on photons and the difference 
between the lines-of-flight of photons and the line joining instantaneous positions 
of source and absorber. 

Since in this laboratory we were undertaking a ' source at centre ' experiment 
similar to that of Hay et al., we decided also to make an experimental test of the 
' peripherally opposite ' arrangement. A null result, besides confirming the 
consensus of theoretical opinion, would give an important check on the absence of 
Doppler effect due to mechanical vibration of the rotor. 

The source of 57 Co in a matrix of 56 Fe, was in the form of a slightly convex foil 
F, 0-001 in. thick, hard-soldered to a short thin steel cylinder C which fitted inside 
the tubular steel rotor (Fig. 1 ) against the rim R ; the absorber was a similar foil of 
natural iron (2% 57 Fe) in the other tip of the rotor. This assembly was spun within 
an evacuated glass vessel provided with a thin window, behind which was a pro- 
portional counter for the 14 kev gamma radiation (Fig. 2). Two standard speeds 
were chosen : 100 rev/sec and 600 rev/sec and in separate experiments two different 
methods were used to restrict the counting of gamma rays to those few degrees of 
the rotor's azimuthal position within which the counter could 'see' the source 
through the absorber. 



248 



Absence of Doppler Shift for Gamma Ray Source 



351 



In the first method, the counts were displayed on the screen of a 100-channel 
kicksorter by a device that caused each gamma ray to deliver to the kicksorter an 
impulse proportional to the azimuth of the rotor at the instant in question ; the 
counts in the four channels corresponding most nearly to the ' seeing ' position 
were totalled. A typical record is shown in Fig. 3. 




Fig. 1. Diagram of the rotor. 



Proportional 
Counter 



Fig. 2. Plan of the rotor, vacuum 
vessel, shielding and counter. 



300 


- '; 


■ i 








i _ 


200 




\ 






1 








_ \ Chosen 
% Four 
' "\ • Channels 




;/ 


/• 








>■ '* 


•' . 








100 




V 1.' 1 

\ i » r 

1 1 











90 180 270 

Azimuthal Position of Rotor (degrees) 



360 



Fig. 3. Plot of typical kicksorter record. The channel number is expressed in terms 
of the angular position of the rotor. 



In the second method, the successive interception of two light beams by one 
arm of the rotor opened and closed an electronic gate ; a scale-of-two device 
prevented the gate from responding to the other arm of the rotor. 

After subtraction of background (measured by placing | in. aluminium over 
the window) the events recorded at 600 rev/sec were found by the first method to 
be (0-4 ±3)% less and by the second method to be (2-0 ± 2-5)% more than those 
at 100 rev/sec ; combining these two results we obtain ( + 0-8 ± 2-0)%. 



249 



352 D. C. Champeney and P. B. Moon 

The width of the resonance was measured for the actual source and absorber 
by the method described by Cordey-Hayes, Dyson and Moon (1960); it was 
thence calculated that a fractional shift h(2v) 2 /c 2 would have given a 9-4% higher 
counting rate at the higher speed. In comparison with this figure, our result 
with its associated probable error may be taken as satisfactory evidence in favour 
of a null result. 

As an incidental check that the null result was not due to instrumental 
deficiencies, an experiment with the source at the centre gave an increase in 
counting rate of (7-1 ±1/7)% at 900 rev/sec in agreement with an expected 
increase of 5-4%. 

References 

Cordey-Hayes, M., Dyson, N. A., and Moon, P. B., 1960, Proc. Phys. Soc, 75, 810. 
Hay, H. J., Schiffer, J. P., Cranshaw, T. E., and Egelstaff, P. A., 1960, Phys. Rev. 

Letters, 4, 165. 
Ruderfer, M., 1960, Phys. Rev. Letters, 5, 191. 

Note added in proof. — It may be pointed out that the experiment is equivalent 
to that proposed by Ruderfer (1960) to test for the effects of other drift. No 
special care was taken to obtain readings in more than one direction, but when 
our results are interpreted according to Ruderfer's ideas they indicate an 
average value for v sin 9, over the six days of experimentation, of 17 ± 43 m sec -1 , 
where v is the magnitude of the component of the ether drift velocity in the 
plane of rotation of the rotor and is the angle between this component and 
the counting axis. 



250 



Volume 4, Number 6 



PHYSICAL REVIEW LETTERS 



March 15, I960 



VARIATION WITH TEMPERATURE OF THE ENERGY OF RECOIL- FREE 
GAMMA RAYS FROM SOLIDS* 

R. V. Pound and G. A. Rebka, Jr. 

Harvard University, Cambridge, Massachusetts 
(Received February 17, 1960) 



The 14.4 -kev y ray emitted without recoil by 
0.1-jLisec Fe 57 in metallic iron 1 " 4 excited great 
interest as the most precisely defined electro- 
magnetic frequency yet discovered. It may be 
adequately well defined to allow measurement of 
the influence of a gravitational potential on fre- 
quency 5 and of other small effects hitherto be- 
yond the sensitivity available in the laboratory. 
As a preliminary step in the operation of an ex- 
perimental system designed to measure the 
gravitational effect, we have been making tests 
to find out whether other influences than the one 
intended might lead to systematic errors by in- 
troducing important frequency shifts not taken 
into account. 

So far the largest such effect found is that of 
temperature. That temperature should influence 
the frequency exactly as we observe is very 
simply explained. Thermally excited vibrations 
cause little broadening through first order 
Doppler effect under the conditions obtaining in 
the solid because the value of any component of 
the nuclear velocity averages very nearly to zero 
over the nuclear lifetime. The precision of the 
y ray of Fe 57 requires the second order Doppler 
effect also to be considered. A shift to lower 
frequency with increased temperature results 
from this because the also well-defined average 
of the square of the velocity of the particle in- 
creases in direct proportion to the average kine- 
tic energy. As a consequence one would expect 
a temperature coefficient of frequency in a 



homogeneous solid, 

(Bv/BT) = -vC /2Mc 2 , 

where C £ is the specific heat of the lattice and 
M is the gram atomic weight of iron. In the high- 
temperature classical limit where C L = ZR, 



(Bv/dT) 



T-oo 



-2.44 xl0" 15 i/ per °K. 



At lower temperatures one would expect a coef- 
ficient reduced by the value of the appropriate 
normalized Debye specific heat function. For 
iron, at 300°K one should find about 0.9 times, 
and at 80°K about 0.3 times, the above classical 
value. 

The temperature dependence has been meas- 
ured by counting the y rays from our 0.4 -curie 
Co 57 source transmitted through enriched Fe 57 
absorbing films (0.6 mg Fe 57 /cm 2 ). The Co 57 of 
the source is distributed in about 3xl0" 5 cm 
thickness below the surface of a 2 -in. diameter 
iron disk, made in the manner described ear- 
lier. 1 Small frequency shifts that result when 
the source and absorber are held at different 
temperatures were measured by using a trans- 
ducer to move the source sinusoidally at ten cps 
toward and away from the absorber at a peak 
speed of about 0.01 cm/sec. A gate pulse and 
mercury relays were used to make one counter 
record during 25 milliseconds of the modulation 
period symmetrically disposed about the time of 
maximum velocity toward the absorber. Another 



274 



251 



Volume 4, Number 6 



PHYSICAL REVIEW LETTERS 



March 15, I960 



counter recorded the corresponding counts with 
the source going away from the absorber. The 
difference of the counts in the two registers 
should be proportional to the relative frequency 
shift of the absorber and source for shifts small 
compared to the line width. Quantitative know- 
ledge of the parameters of the system that are 
involved in determining the constant of propor- 
tionality is rendered unnecessary by adding 
through a clock-driven hydraulic system a con- 
tinuous relative motion of 6.3 xlO" 4 cm/sec di- 
rected oppositely during each of the two halves 
of the time for a given datum point. In this way 
the sensitivity to frequency shift originating in 
the Doppler effect is measured simultaneously 
with the shift sought. The algebraic sum of the 
counting rate differences for the two halves of 
the run are proportional to the shift and the dif- 
ference to the sensitivity. 

The shift at liquid nitrogen relative to room 
temperature is comparable to the line width and 
for that point the two counting rates were re- 
corded at a series of values of the sinusoidal 
modulation amplitude. From these a value of the 
shift and of the apparent line width could be ob- 
tained although difficulties of calibration under 
the conditions of operation have contributed 
strongly to the uncertainties. There is evidence 
that the line appears to broaden with such a tem- 
perature difference by perhaps a factor of 2.3 
which might be evidence that the hyperf ine struc - 
ture splittings are temperature sensitive to 
some extent, as must be expected. 

The data are plotted in Fig. 1. A solid line re- 
presenting the effect expected with a Debye tem- 
perature of 420°K is also drawn. The agreement 
can be regarded as an experimental demonstra- 
tion of the second order Doppler effect using 
thermal velocities rather than a centrifuge. It 
might be remarked that crystalline anisotropy 
might make this source of high velocities useful 
for experiments to the end of detecting such spa- 
tial anisotropies as might accompany ether drift 
or an inertial frame. 

The temperature sensitivity at room tempera- 
ture [experimentally (-2.09 ± 0.24) xlO" 15 per 
degree C, theoretically -2.21 xlO" 15 per degree 
C] is highly relevant to the interpretation of data 




TEMPERATURE 



200 300 

DEGREES KELVIN 



FIG. 1. Fractional shift of energy of 14.4-kev 
gamma-ray absorption of Fe 57 vs absolute tempera- 
ture of the metal. The solid line is derived from as- 
suming a Debye temperature of 420°K. 



from our experiment on the effect of gravitational 
potential. A temperature difference of 1°C be- 
tween the top and the bottom of our 22- meter 
tower would result in a shift about equal to that 
predicted by the principle of equivalence. For 
smaller height differences correspondingly 
smaller temperature differences would be re- 
quired. It is now clear that correction for or 
control of the temperature difference and perhaps 
other parameters must be included in the instru- 
mentation of experiments intending to utilize the 
extreme frequency discrimination available with 
gamma rays of this type. 



*Supported in part by the joint program of the Office 
of Naval Research and the U. S. Atomic Energy Com- 
mission. 

1 R. V. Pound and G. A. Rebka, Jr., Phys. Rev 
Letters 3, 554 (1959). 

2 J. P. Schiffer and W. Marshall, Phys. Rev. 
Letters 3, 556 (1959). 

3 S. S. Hanna, J. Heberle, C. Littlejohn, G. J. 
Perlow, R. S. Preston, and D. H. Vincent, Phys. 
Rev. Letters 4, 28(1960). 

4 G. DePasquali, H. Frauenfelder, S. Margulies, 
and R. N. Peacock, Phys. Rev. Letters 4, 71 (1960). 

5 R. V. Pound and G. A. Rebka, Jr., Phys. Rev. 
Letters 3, 439 (1959). 



275 



252 

TEMPERATURE -DEPENDENT SHIFT OF y RAYS EMITTED BY A SOLID 

B. D. Josephson 

Trinity College, Cambridge, England 

(Received March 11, 1960) 



Recent experiments by MOssbauer 1 have shown 
that when low -energy y rays are emitted from 
nuclei in a solid a certain proportion of them are 
unaffected by the Doppler effect. It is the pur- 
pose of this Letter to show that they are never- 
theless subject to a temperature -dependent shift 
to lower energy which can be attributed to the 
relativistic time dilatation caused by the motion 
of the nuclei. 

Let us regard the solid as a system of inter- 
acting atoms with the Hamiltonian 



#=£^.72™. + ^, 



The MGssbauer effect is due to those processes 
in which the phonon occupation numbers do not 
change. It might appear that in such cases the 
energy of the solid is unaltered, but this is not 
so, as the nucleus which emits the y ray changes 
its mass, and this affects the lattice vibrations. 
Suppose the nucleus of the z'th atom emits a y ray 
of energy E, its mass changing by 6m j = -E/c 2 . 

The change in energy, 6E, of the solid is given 
by 

6£ = < Aff> = 6(p 2 /2m .) = -6w .</> . 2 /2m 2 > 



(6w./m.)7\ 



(B/t 



'.c*)T., 
i i 



where Tj is the expectation value of the kinetic 
energy of the rth atom. The energy of the y ray 
must accordingly be reduced by 6E so there is a 
shift of relative magnitude 6E/E =T i /wjC 2 . The 



same formula can be deduced by regarding the 
shift as due to a relativistic time dilatation. 

To estimate T i we make the following assump- 
tions: (i) The atoms all have the same mass, 
and the kinetic energy is equally distributed 
among them, (ii) The kinetic energy is half the 
total lattice energy, i.e., we assume that the 
forces coupling the atoms are harmonic. Under 
these assumptions 7\-/m ; - = \U, where U is the 
lattice energy per unit mass. The relative shift 
is thus given by 6E/E =£//2c 2 . For Fe at 300°K 
this has the value 8xl0" 13 . Clearly a compen- 
sating shift would occur for absorption provided 
source and absorber were identical and at the 
same temperature. A small difference in tem- 
perature between source and absorber leads to 
a relative shift per degree given by 6E /E =Cp/2c 2 
where Cp is the specific heat. For Fe at 300°K 
this is 2.2xlO" 15 /°K. This is sufficient for it to 
be necessary to take it into account in accurate 
experiments using the resonance absorption of 
y rays, such as those to measure the gravitation- 
al red shift. 2 ' 3 

I would like to thank Dr. Ziman, Professor 
O. R. Frisch, and Dr. W. Marshall for helpful 
discussions. 



'R. L. MSssbauer, Z. Physik 151, 124(1958). 

2 R. V. Pound and G. A. Rebka, Phys. Rev. Letters 
3, 554 (1959). 

3 T. E. Cranshaw, J. P. Schiffer, and A. B. White- 
head, Phys. Rev. Letters 4, 163(1960). 



342 



253 



Volume 5, Number 1 



PHYSICAL REVIEW LETTERS 



July 1, I960 



ACOUSTICALLY MODULATED y RAYS FROM Fe" 

S. L. Ruby and D. I. Bolef 

Westinghouse Electric Corporation, Pittsburgh, Pennsylvania 

(Received June 13, 1960) 



The relationship between the emission of y rays 
by nuclei bound in a crystal and the creation (or 
destruction) of phonons has been discussed by 
Visscher, 1 and suggests that a careful study of 
the "off- resonance" line shape in a Mossbauer- 
type experiment may be used to observe the fre- 
quency distribution of lattice vibrations in the 
crystal. Unfortunately, a direct attempt at such 
a study seems difficult since it requires the 
measurement of nuclear y-ray absorption cross 
sections much smaller than the photoelectric 
cross sections for the same atom. In an attempt 



to investigate the interactions between phonons 
and emitting nuclei, therefore, it was decided to 
generate low-energy phonons acoustically, and 
to study their effect on the y-ray spectrum. 

Source and absorber were one-mil thick 321 
stainless steel (18% chromium, 8% nickel) foils. 
The source, into which had been diffused Co", 
could be driven by either or both of two methods: 
(1) a low-frequency (15 cps) drive utilizing a 
loud speaker, and (2) a piezoelectric quartz 
crystal drive mounted on the rear of the source 
foil. The quartz crystal is driven by a radio- 



254 



Volume 5, Number 1 



PHYSICAL REVIEW LETTERS 



July 1, 1960 



frequency oscillator whose frequency and ampli- 
tude are continuously adjustable. The counting 
rate for the 14.4-kev y ray, as a function of loud- 
speaker velocity, is measured by using the out- 
put of a single -channel analyzer to "command" a 
multichannel analyzer to measure the velocity at 
a particular instant. This is accomplished by 
feeding the amplified output of the velocity pickup 
coil (rigidly attached to the source) to the appro- 
priate place in the analog -to -digital converter 
section of the analyzer. 

The experiment was planned on the assumption 
that the density of ultrasonic phonons in a narrow 
frequency band could be markedly increased over 
that corresponding to 300°K, and that this should 
lead to pairs of satellite peaks, symmetrically 
located relative to the main Mossbauer peak, 
with a spacing in energy units of AE =hq or, in 
velocity units, v s =(c/E )AE. (q/2v = ultrasonic 
frequency, £ = 14.4 kev.) This corresponds to 
the creation or destruction of acoustic phonons 
with the emission of the y ray. A similar dis- 
crete Doppler effect has been observed in optical 
light diffracted by acoustic "gratings. " The opti- 
cal effect is extremely small, of the order of 
10" 4 A for an optical wavelength of 5460 A and an 
ultrasonic frequency of 10 Mc/sec. The effect 
has been observed for both traveling and station- 
ary sound waves. 2 A theory of this optical effect 
has been given by Raman and Nagendra Nath. 3 

Since the source foil is thin (approximately one- 
tenth the wavelength of sound at 20 Mc/sec), one 
can alternatively consider the quartz transducer 
as simply vibrating the foil with a sinusoidal 
velocity, v s =v m cosqt. This corresponds merely 
to a sinusoidal motion of the center of mass of 
the foil. The "instantaneous" frequency of the 
14.4-kev y ray may therefore be expressed as 



v= i/j+Av siaqt, 



(1) 



where 



v =E /h = 3.48x10" cycles/sec, 

and the maximum frequency deviation is Av 
= v (v m /c). Expressed in the language of fre- 
quency modulation, this corresponds to a carrier 
of frequency v , modulated sinusoidally at a fre- 
quency q/2-n. The modulation index is m= {v m /c) 
x(2rtv /q). The frequency spectrum can be shown 4 
to consist of the carrier and an infinite set of 
side -bands, with the nth side frequency sepa- 
rated from the (n + l)th side frequency by the 
modulating frequency, <?/2jr. The maximum am- 
plitude of the nth side frequency is given by 



J n 2 (m), where J n (m) is the Bessel function of the 
first kind of order n. 

In Fig. 1 the solid lines in curves a-e show the 
result of calculations for five values of m, with 
q/2v = 20 Mc/sec. The vertical scale of the draw- 
ing is based on curve a. Also shown in the figure 
are the experimental results, using a 20-Mc/sec 




-2-10 1 2 
VELOCITY, MM /SEC 



50 40 30 20 K> 10 20 30 40 90 
FREQUENCY, MC/SEC 

FIG. 1. Mossbauer pattern for Fe 87 y ray emitted 
by a stainless steel source driven by a 20-Mc/sec 
x-cut quartz transducer. The experimental points are 
shown in a -e for values of the driving voltage, V, 
from to 2.0 volts rms. The solid curves are cal- 
culated on the basis of FM theory, using a single pro- 
portionality constant between m and V rms which best 
fits the data. 



6 



255 



Volume 5, Number 1 



PHYSICAL REVIEW LETTERS 



July 1, I960 



x-cut quartz transducer, taken at five different 
transducer driving voltages. The maximum ve- 
locity of the iron atoms resulting from the ultra- 
sonic vibration has not been measured directly, 
but is expected to be proportional to the driving 
voltage; the proportionality constant has been 
chosen so as to fit the solid curves as well as 
possible to the experimental points. Using 
m = 1.1 V" rms , one finds v m = 0.29 cm/sec for 
1.5 V Tms across the transducer. This value for 
v m is consistent with the value calculated from the 
piezoelectric properties of the quartz transducer. 
The velocity at 20 Mc/sec corresponds to the 
rather small maximum displacement of approxi- 
mately 2xl0" 9 cm. 

The progressive disagreement between the cal- 
culated and experimental curves with higher 
drive voltage, especially near the carrier fre- 
quency, suggests that all of the iron atoms did 
not have the same velocities. A new source foil 
was then prepared, care being taken to preserve 
flatness of the foil and uniformity of the acoustic 
bond. In Fig. 2, a plot of the amplitude of the 
carrier (unshifted y ray) vs the 20-Mc/sec driv- 
ing voltage is given, together with a plot of 
[1 - 0.24J 2 (w)] with m =0.6 V rms . The calcu- 
lated curve assumes that all the Fe 57 atoms have 
the same maximum velocity v m . The pattern 
using the new foil, however, still suggests a con- 
tinuous range of velocities with perhaps 50% of 
the Fe 57 nuclei moving considerably more slowly 
than the remainder. Such an effect could be 
caused by bonding defects, such as air bubbles 
trapped in the cement between foil and quartz. 
Velocity blurring also results from the fact that 
the thickness of the foil is not negligible com- 
pared to the wavelength of the sound waves. 

Since the energy shift of the y ray is deter- 
mined solely by the frequency of the ultrasonic 
drive, this discrete Doppler technique offers a 
precise method for adding or subtracting known 
quantities of energy to the y ray. This may be 
useful in providing a monochromatic calibration 
of energy or velocity in the measurement of line 
splittings (such as Fe 57 in iron) or line shifting 



£ .95 




/^ ^ 


< 






ec 






o 90 

z 

Z.85 




/ ' •• * * 


3 






O 




/" . . 


° MO 


•/ 




u 






> 












•- .75 






< 


• 




_i 






S .70 


1 


1 1 1 1 1 



FIG. 2. Relative intensity of carrier (unshifted y 
ray) vs voltage on quartz transducer. The solid 
line represents the theoretically predicted function 
[l-0.24J B 2 (w)], wherew = 0.6F rms for this case. 
This prediction assumes that all the Fe 57 atoms have 
the same maximum velocity v m . 



(such as temperature shifts of the Mossbauer 
peak due to zero-pomt vibration). This method 
for Doppler shifting may also be applicable at 
low temperatures when more conventional drives 
are inconvenient to use. For this purpose, broad- 
banding of the transducer frequency response will 
be desirable. 

We wish to thank Dr. L. Epstein for help in the 
preparation of the source, Mr. John Hicks for 
his careful and ingenious help throughout the ex- 
periment, and Dr. Meir Menes, who first sug- 
gested the FM approach. 



'W. M. Visscher, Ann. Phys. 9, 194 (1960). 

2 For a review of this work see L. Bergmann, Ultra- 
sonics (John Wiley & Sons, New York, 1951), pp. 66 ff. 

3 C. V. Raman and N. S. Nagendra Nath, Proc. 
Indian Acad. Sci. (A}2, 406, 413 (1935); and (A) 3 , 75 
(1936). 

4 S. Goldman, Frequency Analysis, Modulation and 
Noise (McGraw-Hill Book Company, New York, 1948). 



256 



PHYSICAL REVIEW 



VOLUME 124, NUMBER 3 



NOVEMBER I , 1961 



Measurement of the Refractive Index of Lucite by Recoilless Resonance Absorption* 

L. Grodzins and E. A. Phillips 

Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 

(Received June 20, 1961) 

A method of frequency-modulating a monochromatic electromagnetic wave by varying the optical path 
length between the source and detector is described. The method has been applied to the measurement of 
the refractive index of Lucite for the 0.86 A radiation emitted from Co 67 ; the small frequency shift was 
detected by recoilless resonance absorption. The refractive index was found to be 1— n= (1.29±0.03)X10 -6 , 
in agreement with classical theory. 



THIS paper describes a method of frequency- 
modulating a monochromatic electromagnetic 
wave by varying the optical path length between the 
source and detector. The method has been applied to, 
and is described in terms of, the measurement of the 
refractive index of Lucite for the 14.4-kev radiation 
emitted from Co 57 . The measured refractive index 
agrees, within the 2% experimental uncertainty, with 
the simple theory applicable when the radiation energy 
is much greater than the binding energy of the electrons 
in the refractive medium, as in this case. The technique 
is in principle applicable to the nearly monochromatic 
radiation emitted from optical-frequency masers. 

It is instructive to consider the method from two 
points of view, first in terms of frequency modulation 
and then in terms of a Doppler shift. Consider a source 
S and an observer (in our case a recoilless resonance 
absorber) A separated by a distance x [Fig. 1(a)]. A 
wave of angular frequency o> emitted by 5 will have the 
form e ia(t ~ xle) at A. If a length L of material with re- 
fractive index n is placed in the optical path, the wave 
becomes e ia< - t ~ x,e)+i ' t ', where the phase advance 



shift 



<f>= (l — n)o}L/c. 



(1) 



If <f> changes with time, the instantaneous frequency 
seen by A will be (o>+d<t>/dt). This is done by moving a 
wedge-shaped piece of material to produce a frequency 

* This work is supported in part through a U. S. Atomic 
Energy Commission contract, by funds provided by the U. S. 
Atomic Energy Commission, the Office of Naval Research, and 
the Air Force Office of Scientific Research. 



1 d<f> 

=Av = 

lirdt 



(1-n) dL 
c it 



(2) 



An equivalent point of view considers the radiation as 
being Doppler-shifted during the refraction by the 
moving wedge [Fig. 1(b)]. As it leaves the wedge the 
radiation is deflected (toward the normal, since «<1) 
by an angle 

A0=(1— n) tana. 

The change in momentum of the photon is Ap=pA6, 
and since the wedge is moving at a speed V it does 
work on the photon, increasing its energy by 

A£= VAp= Vp{X-n) tana=£[(l-«)/c]7 tana, 

which is equivalent to Eq. (2) above. 

For 14.4-kev radiation, the refractive index of Lucite 
is (see below) 

(1-») = 1.29X10- 6 , 
so that 

{Av/v) Xi kev =4X lO~ 17 dL/dL 

The frequency shift thus obtained for reasonable values 
of dL/dt can be detected by recoilless resonance 
scattering. 1 

A schematic drawing of the experimental arrange- 
ment is shown in Fig. 2(a). The recoilless resonance 



■R. L. Mossbauer, Z. Physik 151, 124 (1958); R. V. Pound 
and G. A. Rebka, Jr., Phys. Rev. Letters 4, 337 (1960). 






257 



MEASUREMENT OF REFRACTIVE INDEX OF LUCITE 



775 



apparatus has been described previously. 2 The 14.4-kev 
gamma rays from a Co 67 source diffused into Armco 
iron passed through a rotating wheel, shown in profile 
in Fig. 2(b), then through a movable 0.5-mil Armco 
Fe absorber to the Be- window Nal(Tl) detector. 

To construct the wheel, 12-in. diam. pieces of YS-in. 
brass and j-in. Lucite were clamped together and 120 
radial slots y§- in. wide were cut through both at an 
angle of 60°. Since the gamma rays are stopped by the 
brass, they are allowed to pass through only one side 
of each Lucite tooth. As the wheel rotates, every gamma 
ray which passes through it does so when the thickness 
of Lucite in the tooth is changing in the same direction. 
Thus all the gamma rays detected undergo a frequency 
shift of the same direction and magnitude. 

The absorption line profiles for four wheel speeds are 
shown in Fig. 3. The shift in the line position is evident. 
The high-speed runs, 1500 rpm clockwise and counter- 
clockwise, show a broader line which we attribute to 
vibration transmitted through the air from the rotating 
wheel to the source and absorber. This effect diminished 
rapidly with decrease of angular speed and no attempt 
was made to alleviate it. At speeds below 1000 rpm, 
line broadening resulted in an error of less than 2% in 
the determination of the line shift. 

The shift in the position of an accurately known line 
profile is most efficiently determined by measuring the 



Phose advance 




Fig. 1. (a) The phase advance produced by interposition of a 
length L of refractive material between source 5 and observer A . 
(b) The deflection A0 of the beam when the refractive material of 
Fig. 1 (a) is wedge-shaped. 




ftMfcg 



Source Wheel Absorber Detector 
(a) 

I Detector 




Fig. 2. (a) Schematic diagram of experimental arrangement, 
(b) Detail of the slotted wheel ; not to scale. 

change in counting rate at the maximum slope points 
of the absorption line profile, 0.013 cm/sec in this case. 
The counting rates for absorber speeds towards and 
away from the source were separately recorded for 
each of a set of speeds of the Lucite wheel ; the resulting 
line shift as a function of wheel speed is shown in Fig. 4. 
A least-squares fit of the data (from 4 to 12X10 6 counts 
per point) between 900 rpm clockwise and 900 rpm 
counterclockwise yields a slope 

Av Av 1 cm/sec 
=c— =(3.32±0.05)X10" 6 , (3) 



Attwheel V Awwheel 



rpm 



where Av/v is the relative frequency shift. 

The effective radius from the axis of the wheel to 
the gamma ray path was 14.2 cm so that 



(dL/dt) 

Awwheel 



2.58- 



cm/sec 



rpm 



(4) 



A combination of Eqs. (2)- (4), yields the refractive 
index : 



(1-n)- 



c{Av/v) 3.32 X10- 6 



(dL/dt) 



2.58 



(1.29±0.04)X10- 6 . 



*L. Grodzins and F. Genovese, Phys. Rev. 121, 228 (1961). 



The stated error includes uncertainties in the effective 
radius, the absorption line depth, and the absorber 
velocity. 

This result is in agreement with the theoretical value 
obtained for the refractive index of a gas in which the 
binding energy of the electrons is much less than the 



258 



776 



L. GRODZINS AND E. A. PHILLIPS 



energy of the radiation. This condition holds for 14.4- 
kev radiation on Lucite (CbH 8 2 ) since Ek (oxygen) 
=0.53 kev. The result is 3 



(l-n) = N<r 



2A nuoh 



where the symbols have their conventional meaning'; 
i.e., No is Avogadro's number, p is the density, etc. 
For the case of Lucite and 14.4-kev radiation, Z/A 



Da*. 

5BK 


J ' 


r— 

t 


- 1 1 1 r ■ 

LUCITE WHEEL 

h 1 


i i i 
* 


57K 






f 


\ i , 


t 
i 










1500 RPM CW 


56K 








- 




> 






» * 




59K 


_ 






- 


58K 
57K 


"{ 


* 


♦ 


1 

♦♦, i H 

t 1 


f ( t" 

500 RPM CW 


56K 








- 


< 


► 






♦ ! ♦ 




69K 








tit 




58K 


-t 


t 




>t 

♦* i / 


500 RPM CCW 


57K 


- 




t 


56K 


_ 






- 




► 






♦♦ t 

1 




59K 
58K 


J 


1 


♦ 


1 l 1 


56K 


- 






lit 


1500 RPM CCW 


S6K 








',; 




54K 
S3K 


1 


l 


, 


» 

i i i 


l l 1 



.06 .06 .04 .02 O .02 .04 .06 .08 .0 
AbMrbtr Velocity cm /sec 

Fig. 3. Absorption line profile versus wheel speed. 




800 1000 1200 



Wheel Sptrt RPM 



* F. K. Richtmeyer and E. H. Kennard, Introduction to Modern 
Physics (McGraw-Hill Book Company, New York, 1947), 4th ed., 
pp. 522-527. 



Fig. 4. Reduced data: Line shift versus wheel speed; the 
theoretical slope contains an uncertainty in the wheel radius. 
(See text.) 



=0.54, p= 1.185, and w=2.185X10 19 . Then 
l-»= 1.285 X10- 6 . 

The exact theory of the index of refraction, which 
takes into account the binding energies of the elec- 
trons, 45 yields a result differing from the above number 
by about 0.1%. 

The measurement of the index of refraction at x-ray 
wavelengths is, of course, not new. 3 Indeed, Bearden* 
measured the refractive index of diamond at 1.39 A to 
an accuracy of 1 part in 10*. The phase modulation 
technique can, if desirable, be made as accurate for 
those wavelengths observed by recoilless nuclear gamma 
emission. 

We have shown that the frequency of a nearly mono- 
chromatic electromagnetic wave may be shifted by 
modulating the optical path between source and ob- 
server. The application to an optical-frequency maser 
where Av/p<lOr* is evident. Since (1— n) is ~10* 
times as large for optical frequencies as for x rays, fre- 
quency modulation may be observed by varying either 
» or L. For example, L may be varied by vibrating a 
mirror [1— »=2 in Eq. (2)] from which the light is 
reflected. The corresponding experiment for recoilless 
gamma radiation has been reported by Ruby and Bolef,' 
who acoustically vibrated the source. 

* J. A. Bearden, Phys. Rev. 54, 698 (1938). 
s J. A. Prins, A. Physik 47, 479 (1928). 

• S. L. Ruby and D. I. Bolef, Phys. Rev. Letters 5, 5 (1960). 



259 



Time Dependence of Resonantly Filtered Gamma Rays from Fe 57 f 

F. J. Lynch, R. E. Holland, and M. Hamermesh 

Argonne National Laboratory, Argonne, Illinois 

(Received June 6, 1960) 



The time dependence of gamma rays emitted by the 14.4-kev state of Fe 57 has been studied by delayed- 
coincidence measurements between a 123-kev gamma ray preceding formation of the state and the 14.4-kev 
gamma ray from the state. When no filter was used, the number of gamma rays decreased exponentially 
with the known half -life of 0.1 /usee. When a foil of Fe 57 (which was resonant to 14.4-kev radiation) was 
used as a filter, the number of gamma rays observed through the filter did not decrease exponentially. 
Instead, the filter absorbed almost none of the gamma rays first emitted by the 14.4-kev state; at later times 
the absorption increased. Data were taken with three different thicknesses of absorber and with emission 
and absorption peaks separated by to 11 times the width of the resonance. The energy separation resulted 
from the Doppler shift associated with a constant velocity between source and absorber. These data were, 
for the most part, in good accord with the prediction of a theory based on a classical model for absorber and 
source. In particular, the results verified the theoretical prediction that at certain times the intensity of 
radiation observed would be greater with the filter than without it. 



I. INTRODUCTION 

IN an earlier paper, 1 we showed that the intensity of 
the gamma rays transmitted through an absorber 
which is resonant to the incident gamma radiation does 
not decrease exponentially with time. Instead, the trans- 
mitted beam appears initially to decay faster than the 
rate corresponding to the life-time r of the emitting 
state. The intuitive picture which led us to undertake 
the initial experiments was that the resonance absorp- 
tion tends to reduce the intensity at the center of the 
emitted line relative to the intensity in the wings. 
Thus the width at half maximum of the remaining peak 
is greater than the value T=H/t of the original peak; 
and the lifetime of the state appears to be correspond- 
ingly shorter. 

A quantitative theoretical treatment was developed 
by describing the emitted radiation as a damped 
electromagnetic wave. 2 On passing through a medium 
filled with resonators, the frequency spectrum of the 
radiation is altered, so that its time dependence is no 
longer exponential. The theoretical analysis, which is 
presented in Sec. II, suggested some of the experiments 
which are described in Sec. III. 

The 14.4-kev state of Fe 57 provides a convenient 
source for observation of this effect. The formation of 
the 14.4-kev state is announced by a 123-kev gamma 
ray in the decay 3 of Co 67 , the Mossbauer effect is large 4 
(about 60% of the radiation is emitted without recoil 
at room temperature), the half -life 3 is 0.1 ^sec, and the 



fWork performed under the auspices of the U. S. Atomic 
Energy Commission. 

1 R. E. Holland, F. J. Lynch, G. J. Perlow, and S. S. Hanna, 
Phys. Rev. Letters 4, 181 (I960). 

*M. Hamermesh, Argonne National Laboratory Report 
ANL-6111, February, 1960 (unpublished), p. 6. 

* Nuclear Data Sheets, National Academy of Sciences, National 
Research Council, 1959 (U. S. Government Printing Office, 
Washington, D. C). 

«R. L. Mossbauer, Z. Physik 151, 124 (1958); J. P. Schiffer 
and W. Marshall, Phys. Rev. Letters 3, 556 (1959); R. V. Pound 
and J. A. Rebka, Jr., Phys. Rev. Letters 3, 554 (1959). 



gamma ray is highly converted 6 (a =15) so that 
imprisonment of the resonance radiation need not be 
considered. 

II. THEORY 

The time dependence of the 14-kev radiation of 
Fe 87 as observed in transmission through an Fe 67 
absorber can be explained on a simple classical theory. 
As in standard treatments of emission and dispersion, 
the medium is assumed to consist of damped oscillators 
with natural frequency wo and decay constant X. The 
radiation emitted by the Fe 67 source has an electric 
field 



a(/) = exprjtw /- 



1 r 

§x/]=— I 

2iri J- 



to— wo— ?*X 



■r 



dto a(to)e^. (1) 



The constant X is just the reciprocal of the mean life 
of the excited state. In the case of the 14-kev line of 
Fe 57 , 93% of the radiation is internally converted; the 
quantity X contains contributions from all types of 
decay (radiation and internal conversion). 

The effect of transmission through the Fe 57 absorber 
can be found by standard methods. 6 Each mono- 
chromatic component a(a))e*"' excites forced oscillations 
of the resonators in the medium. The complex dielectric 
constant is 



e(w) = l+r^-^+wX)" 1 



(2) 



We have written coo' for the resonant frequency of the 
absorber to take account of a possible Doppler shift 
due to motion relative to the source. 



5 H. R. Lemmer, O. J. A. Segaert, and M. A. Grace, Proc. 
Phys. Soc. (London) A68, 701 (1955). 

•J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book 
Company, Inc., New York, 1941), p. 321. 



513 



260 



514 



LYNCH, HOLLAND, AND HAMERMESH 



From Eq. (2), the propagation vector in the medium 
is found to be 

k= (w/c^l+Kwo^-c^+wX)- 1 ]*. (3) 

The wavelength of the radiation is about 10~ 8 cm and, 
as in the x-ray case, we can expand Eq. (3) and retain 
only the first term. The effect of passage through the 
absorber is to change a(w) to a'(w), where 

a'(«) = c(«) exp{-2^[w ' 2 -w 2 +wX]- 1 }, (4) 

in which b is a constant. 

At w=w ', a'(wo') = a(w ') exp[— 2b/\~], so the trans- 
mission at the center of the line is exp[— lb/\\. 

By combining Eqs. (1) and (4), the time dependence 
of the transmitted amplitude is found to be 



1 c 

*'«)=— I 

2nd J_ 



dor- 



>— Wo - §*X 



[2iboi "I 
; — • 
w 2 — wo' 2 — *wXj 



(5) 







v ' aJ-o ' 

>v »I6 


- 


io-' 




rv^o 


- 


lO" 2 


:zi 


V 


\ - 






r 


V 75%' 


io" 3 — 


/ 


\ = 




- 




100% - 


Iff 4 




, 11 , 


1 



This integral is evaluated in the Appendix with the 
result expressed as either 

o'(t) = exp\jo)o't— %\Q 

<*> ri{w — wo'} 1" 
X E (bt)**J n (2bW), (6a) 

n=oL b J 



12 3 4 

t/T 

Fig. 2. Time dependence of radiation after transmission through 
a resonant filter according to Eq. (7), assuming all radiation is 
recoilless or 75% is recoilless 03=16, Ao>=0). The straight line 
represents an exponential decay for comparison. 



: exp iojo't — 1\\ — expil h(w — w ')2 

I 2JI Lw -w ' J 

+ £ (— ) (iO-»'V„(2ftn*)l. (6b) 



'(ty 



The time dependence of the transmitted intensity is 
given by |o'(/)| 2 . Setting w — wo'=Aw, /3=46/X, and 



io-'- 



T=\t in Eq. (6a) yields 

4 Ao>1 B f jSr-]*" 

i-- 

In particular, for Aw=0 this reduces to 
| a '(r)p=e-^/ 03iri)] 2 . 



I oo r 4 Awyrjer-ii" 

I «=o L jS X J L 4 J 



(7) 



(6) 



For large values of Aw, the series (7) converges very 
slowly. In this region, we use Eq. (6b) for a'(t). The 
transmitted intensity is 



XV ' 


1 

Aw 


1 
»0 
= 4 


- 


- v\ 






- 








. 


= \ 






\ 75% ~ 









100% 


1 






1 



l«'(DI ! 



■expt 



+ E 



fAw X /Si 

— T+ 

L X Aw 4J 



JntfW) 



(9) 



IO" 2 



io- 3 - 



12 3 4 

t/T 

Fig. 1. Time dependence of radiation after transmission through 
a resonant filter according to Eq. (7), assuming all radiation is 
recoilless or 75% is recoilless 03=4, Ao>=0). The straight line 
represents an exponential decay for comparison. 



In computing the time behavior of the transmitted 
intensity, Eq. (7) converges rapidly for 

2 Aw 

— Gsr)*<i, 

P X 
while Eq. (9) gives rapid convergence for 
2 Aw 



■0ST)*>1. 
fi X 

The detailed comparison of the theoretical formulas 
with the experimental results is quite complicated. 
Here we only indicate some of the points to be 
considered. 

Only some fraction of the 14-kev radiation is recoil- 



261 



TIME DEPENDENCE OF RESONANTLY FILTERED y RAYS 



515 



less, while the remainder is shifted far from resonance 
and decays exponentially. In Figs. 1 and 2 we show the 
time dependence of the transmitted beam for two 
different absorber thicknesses when Aw=0, for the 
pure resonance radiation and for the case when 75% 
of the radiation is recoilless. 

The effect of relative motion of source and absorber 
on the time dependence of the transmitted beam is 
shown in Fig. 3 (Aw=5X) and Fig. 4 (Aw=4X). 

Figure 4 shows the surprising behavior which should 
occur as the source and absorber frequency are separated 
more and more: the intensity oscillates about the 
exponential curve. Thus, at certain times more gamma 
rays are received through the absorber than would have 
arrived if the absorber were absent. The medium 
behaves like a resonant filter and appears to "ring" in 
response to the incident damped oscillation. As Aw is 
increased the oscillations shown in Fig. 4 are shifted 
toward shorter and shorter times so that, for very large 
Aw, the normal exponential decay is approached. 

The transmitted intensity given by Eq. (7) or (9) 
can be written as 

|a'(*)| 2 =exp(-X*)FG3,*), 

where X= 1/r. The deviations from exponential behavior 
are more easily seen if one plots the product of the 
counting rate with the factor expQu). For the emitted 
line this product is a constant. For the transmitted line, 
the theory predicts that the product will vary as F(J3,t). 
The theory given above assumes that there is a 
single emission (and absorption) line. However, the 
14.4-kev line of Fe 87 has a hyperfine structure of six 
lines. 7 Since the separation of the hyperfine components 
is large compared to the linewidth, we assume that 



Fig. 3. Effect of 
relative motion on 
time dependence of 
transmitted radia- 
tion according to 
Eq. (7) 03 = 4, Ato 
= JX). For compari- 
son, the straight line 
represents an expo- 
nential decay. 



Fig. 4. Effect of 
relative motion on 
time dependence of 
transmitted radia- 
tion according to 
Eq. (7) 03 = 4, Ao> 
=4X). For compari- 
son, the straight line 
represents an expo- 
nential decay. 




DELAY 
IOOO FT 
RG 63/U 



HP 202 A 
FUNCTION 
GENERATOR 



3L 



AMPLIFIER | 



| HP 460 A | 



T 



I HP 460 B | |HP460A| 



[COINCIDENCE I 



ATE *]_ 



Fig. 5. Schematic diagram of equipment. 

each emitted hyperfine component is absorbed only in 
the corresponding transition in the absorber. By use 
of the values for the intensities of the hyperfine compo- 
nents, the time dependence of the transmitted radiation 
is found to be 

|a'(0|»=exp(-XO[l-/+/{iF(fcM) 

+mw)+m-hmi (io) 

where / is the fraction of gamma rays emitted without 
recoil and /? is the thickness of the absorber expressed in 
mean free paths at the peak of the absorption curve. 
Although in principle /3 is measurable, it is difficult to 
determine it accurately. We have used £ as a parameter 
in fitting the theoretical formula to the experimental 
data. Equation (10) was evaluated on an IBM-704 
computer for various values of /? and the other param- 
eters, and the calculated curve giving the best visual 
fit was plotted with the data. One should expect that /3 
would be given by N<r f, where /' is the fraction of 
Fe 67 nuclei which can absorb without recoil, N is the 
number of atoms of Fe 67 per cm 2 , <7 =47rX 2 / (1+a), X 
is the wavelength of the 14.4-kev gamma ray divided 
by 2ir, and a is the internal conversion coefficient. 

IH. EQUIPMENT AND PROCEDURE 

The measurements were made with a source of Co 67 
(25 000 disintegrations of Co 57 per sec) co-plated 8 with 
Fe 66 on a thin copper foil and annealed in vacuum at 
800°C. Absorbers consisted of rolled foils of Fe 67 
(enriched to 75% in Fe 57 ) or normal Armco iron rolled 
foils annealed at 800°C. The geometric arrangement of 
source, absorber, and detectors is shown schematically 
in Fig. 5. The source, f in. in diameter, was mounted 
on an extension of the speaker diaphragm which could 
be used to shift the resonant frequency of the source by 
the Doppler effect. The absorber foil (1 in. in diameter) 
was clamped between two Lucite disks mounted f in. 
above the source. The detector for the 14.4-kev gamma 



7 S. S. Hanna, J. Heberle, C. Littlejohn, G. J. Perlow, R. S. 
Preston, and D. H. Vincent, Phys. Rev. Letters 4, 177 (1960). 



8 S. S. Hanna, J. Heberle, C. Littlejohn, G. J. Perlow, R. S. 
Preston, and D. H. Vincent, Phys. Rev. Letters 4, 28 (1960). 



262 



516 



LYNCH, HOLLAND, AND HAMERMESH 



ray consisted of a Nal(Tl) scintillator 1 in. in diameter 
and 0.006 in. thick mounted on a Lucite light pipe 
and an RCA 7265 photomultiplier tube. A second 
Nal(Tl) scintillator, 1 in. in diameter and \ in. thick, 
served as a detector for the 123-kev gamma ray and 
was mounted § in. from the source on a line making an 
angle of 120° with the line connecting the centers of 
the other detector and the source. 

The speaker coil was driven with a peak-to-peak 
amplitude of \ in. by the amplified signal coming 
alternately from a triangular wave generator 9 and 
from a 40-cps sinusoidal signal. The triangular wave 
caused the source to move with a constant velocity 
either toward or away from the absorber, except during 
the short interval of reversal of direction. This constant 
velocity produced a small constant increase or decrese 
in gamma-ray energy through the Doppler effect. On 
the other hand, the sinusoidal signal, because of its 
much higher frequency, produced a wide range of 
energy shifts; most of the time the gamma rays were 
not resonant nor even nearly resonant with the absorber. 
Other sources of absorption, such as the photoelectric 
effect, would be scarcely affected by the energy shift. 
Thus we observed the effect of both nonresonant and 
resonant absorption for small energy shifts with the 
triangular wave applied to the voice coil and we 
observed the effect of nonresonant absorption alone 
with the sinusoidal voltage applied to the voice coil. 

Because the average geometric arrangement was 
slightly different for the two signals, measurements 
were taken without an absorber, and the sinusoidal 
signal was adjusted slightly to bring the coincidence 
counting rate within 1% of the counting rate with the 
triangular drive. 

It is desirable that the triangular wave of voltage 
should move the source with constant speed. The 
degree to which this condition holds is determined by 
the linearity of speaker movement with current, the 
linearity of the triangular wave voltage, the duration 
of the transient vibration occurring during reversal of 
the direction of motion, and the isolation of the system 
from mechanical vibrations conducted by the air or 
by the building. 

We investigated the movement of the voice coil as a 
function of current and found the relationship to be 
linear within the accuracy of measurement (2%). 
The duration of the transient at the peaks of the 
triangular wave from the generator was known to be 
small. From the observed duration of the transient 
voltage induced by a sudden displacement of the voice 
coil, the effect due to the inertia of the voice coil and 
source holder was estimated to be less than 25 msec. 
At the frequencies we used (<0.2 cps), this corre- 
sponded to less than 1% of the period. 

The transmission of noise to the speaker was greatly 
reduced by mounting the assembly consisting of 

•Model 202A function generator, manufactured by Hewlett- 
Packard Company, Palo Alto, California. 



speaker, source, absorber, and detectors in a box lined 
with sound-absorbing material and acoustically isolated 
from the floor. With these precautions, the observed 
vibration of the voice coil because of acoustic noise 
was less than 0.0004 in. per sec, which should be 
compared to the velocity of 0.0037 in. per sec needed to 
produce a Doppler shift of one resonance width. The 
over-all performance of the system was investigated by 
observing the width of the central dip in transmission. 
This was 20% wider than expected for the thickest 
absorber and 40% wider for the thinnest. 

The circuits for measuring the coincidence rate as a 
function of time delay after formation of the 14.4-kev 
state are shown schematically in Fig. 5 and were the 
same (except for slight modifications) as those used 
previously to measure the lifetimes of excited states of 
nuclei. 10 The output of the time-to-pulse-height con- 
verter was stored in the right half of the 256-channel 
analyzer when the voice coil was driven with the 
triangular wave and in the left half when the voice 
coil was driven with the sinusoidal voltage. During a 
run, the triangular wave and sinusoidal voltages were 
applied alternately for 4-min intervals by a timing 
mechanism, and data were accumulated over a 24-hr 
period. Signals from slow amplifiers and single-channel 
analyzers set on the photopeaks of the two gamma rays 
were required in order to record an event. 

Because of the method of recording data, slow 
drifts were not important. The calibration of the 
time-to-pulse-height converter (as obtained by using 
the data from the sinusoidal run to measure the 
lifetime of the 14.4-kev state of Fe 57 ) remained constant 
within 1.5% over a period of 1 month. A slow drift of 
the peak channel was also observed (about \ channel 
per day). The converter was linear to within 1% in the 
region of interest. 

IV. RESULTS AND DISCUSSION 

The results of a typical measurement for the thickest 
absorber foil (2.7 mg/cm 2 of Fe 67 , isotopic enrichment 
75%) is shown in Fig. 6. The upper part of the figure 
shows a conventional semilogarithmic plot of the data 
after subtraction of background due to accidental 
coincidences. This background was determined from 
the counting rate at times preceding the peak shown in 
Fig. 6(a). In general it was less for the part of the run 
in which the source was moved at constant velocity 
than for the part in which the source was vibrated 
sinusoidally because the counting rate in the 14.4-kev 
detector was less in the former condition. The total 
number of accidental coincidences was 2.5% of the 
total number of true coincidences for the data with 
vibrated source shown in Fig. 6. The steep rise at the 
left side of the figure represents the time resolution of 
the fast circuit; it corresponds to a resolution curve 



» F. J. Lynch and R. E. Holland, Phys. Rev. 114, 825 (1959); 
R. E. Holland and F. J. Lynch, Phys. Rev. 113, 903 (1959). 



263 



TIME DEPENDENCE OF RESONANTLY FILTERED 



RAYS 



517 



with a full width at half maximum of 25 rmisec. The 
curve obtained with vibrated source shows the expo- 
nential decay with a half-life of 0.10 /usee in agreement 
with previous measurements. 3 The curve obtained with 
the stationary source demonstrates the effect we 
expected. 

The lower portion of Fig. 6 shows the same data 
plotted in such a way as to exhibit the deviations from 
exponential decay predicted in Sec. II. Here the 
count in each time channel has been multiplied by e*', 
where X is the decay constant and t is the time, and the 
resulting numbers have been adjusted so that the 
average value of the vibrated data was equal to unity 
after the initial rise. Data in several adjacent channels 
were averaged when the counting rate was low. Vertical 
bars give the standard deviations as calculated from 
the number of counts. The solid curve shown was 
calculated for this absorber from the prescription given 
at the end of Sec. II, with the thickness parameter 
/3 adjusted for best fit to the data. This value of /J was 
twice that expected from the thickness of the absorber. 

Figure 7 gives typical results for a number of runs 
in which the source was moved at constant velocity 
alternately toward and away from the absorber, and 



•. % VIBRATED 



STATIONARY 



v.' 



*vi: 




■o o0 5 i ■ 
VIBRATED 



it, 



**U-.-LU 



STATIONARY 



20 40 60 80 

TIME CHANNEL ( I CHANNEL »9.4 m/i$«c) 

Fig. 6. (a) Semilogarithmic plQt of delayed time spectra taken 
with source and absorber (2.7 mg/cm s of Fe", isotopic abundance 
75%) stationary and with source and absorber vibrated relative 
to one another; (b) Data of Fig. 6(a) replotted with ordinate 
multiplied by « X| in order to compare with theoretical expression 
given by the solid line. 



data were accumulated without distinguishing the 
direction of motion. This motion of the source produced 




1.0 
0.8 
0.6 
0.4 
0.2 

oU 



I A I I I I I I I I 



AE = ur 



10 20 30 40 50 60 70 80 
TIME CHANNEL (I CHANNEL = 9.4 m^sec) 

Fig. 7. Time spectra obtained with various energy shifts of the 
emitted gamma ray. The absorber was the same as in Fig. 6. 
The energy shift AE is given in terms of the linewidth deduced 
from the mean life of the state, T=h/r. The dashed curves give 
the time spectra when the source is vibrated; the solid curves 
are the theoretical predictions. 



264 



518 



LYNCH, HOLLAND, AND H A M E R M E S H 



a shift in energy of the emitted gamma ray. The 
amount of the shift, AE, is given on the figure in terms 



1.0 
0.8 


. ' 1 " '___'_ ■ ' 1 .' 


■L 


. 


0.6 
0.4 


"i 


.'ilp 


0.2 


.f AE«0 


- 


Ot 





1 1 1 


i.o- 


-i 1 1 1 1 1 — 


1 1 ■ 1 


0.8- 
0.6- 




0.4- 




- 


0.2- 


! AE = ir 


- 


Qi 




1 1 1 


i.o- 


i i ■ >-\ ' — i — ' — 
Ls&togMi+j 


1 1 1 - 


0.8- 
z 
8 0.6- 


"pv/ 




- 0.4- 
0.2- 


j AE=3r 

i 








i.o- 


^^^^i^ 


-l+ift 


0.8- 


V 




0.6- 




- 


0.4- 




- 


0.2- 


j AE«5r 


- 


0^ 


' 


1 1 1 




i.o- 


J^j&^UiJ. 


-M+U 


0.8- 


H 




0.6- 


f 

• 


- 


0.4- 
0.2- 


1 

f AE=7r 

1 


; 


0* 


4 . , . 1 1 


1 1 1. 



10 20 30 40 50 60 70 80 
TIME CHANNEL (I CHANNEL -9.4 m/tsecj 

Fig. 8. Time spectra obtained with various energy shifts of the 
emitted gamma ray. The absorber was a rolled foil (1.27 mg/cm 2 
of Fe", isotopic abundance 75%). The energy shift AE is given 
in terms of the line width deduced from the mean life of the state, 
T=A/t. The dashed curves give the time spectra when the source 
was vibrated; the solid curves are the theoretical predictions. 



of T=h/r, the expected full width at half maximum of 
the emitted line. The data taken with the source 
vibrated has been represented in these figures by the 
light dashed line in order to keep the number of data 
points from becoming so large as to be confusing. The 
solid curve is as before a theoretical curve calculated 
according to Sec. II, with the foil thickness parameter 
the same as that used in Fig. 2. Similar data for a 
thinner foil (1.27 mg/cm 2 of Fe 67 in an enriched foil) 
is shown in Fig. 8 and for a still thinner foil (0.22 mg/cm 2 
of Fe 67 in a normal isotopic concentration) in Fig. 9. 

Not shown are data which were obtained to verify 
the prediction of the theory that the delayed time 
spectrum depends only on the magnitude of the shift in 
gamma-ray energy and not on its sign. A run was made 
in which data were accumulated only during travel of 
the source in one direction and the result was compared 
with that obtained when data were accumulated during 
travel in both directions. No difference could be 
observed between the two runs. 

All of the data have been compared to the theoretical 




10 20 30 40 50 60 70 80 
TIME CHANNEL (I CHANNEL = 9.4 m/tsec) 

Fig. 9. Time spectra obtained with various energy shifts of the 
emitted gamma ray. Absorber was rolled foil (0.20 mg/cm J of 
Fe", natural isotopic abundance). The energy shift AE is given in 
terms of the linewidth deduced from the mean life of the state, 
T=H/t. The dashed curves give the time spectra when the source 
was vibrated ; the solid curves are the theoretical predictions. 



265 



TIME DEPENDENCE OF RESONANTLY FILTERED y RAYS 



519 



expression at the end of Sec. II, the values of parameters 
given in Table I being used. These are all as expected 
except j8; all theoretical curves are plotted for a value 
of /8 twice that obtained from the weight of the absorber 
foil. With this reservation and when allowance is made 
for the effect of finite resolution time, the fit between 
theory and experiment is fairly good. In particular, 
the predicted overshoot was observed; this is most 
noticeable in Fig. 7, where the curves for 5r, 7r, and 
lir show that, at certain times, a higher intensity is 
obtained with the absorber than without it. 

We have chosen to plot the relative transmission as 
a function of time, using the energy shift AE as a 
parameter to label the various curves. This has the 
advantage of allowing a direct estimate of the effect 
of time resolution. One could, of course, plot the trans- 
mission as a function of AE with the time as a param- 
eter. This is done in Fig. 10 which shows the theoretical 
and experimental transmission (relative to that of the 
vibrated condition) vs AE for the times t=r/2 and 
/=4r after the formation of the excited state. Note 
that the apparent half-width at half maximum decreases 
from about 3r at t=r/2 to about 0.7r at /=4r. The 
half-width measured without consideration of time was 
about 1.6r for this particular absorber (1.27 mg/cm 2 
Fe»). 

In some cases the discrepancies are larger than 
experimental error. A number of possible sources of 
deviation from the theory were investigated. First, 
the theory assumes that no scattered radiation was 
detected. However, the geometric arrangement we 
used in order to keep the coincidence rate high was such 
that any appreciable scattering would have been 
detected. We made a crude check of this by increasing 
the distance from the source to detector from f in. to 
2 in. The fact that no change in the time spectra other 
than the reduced counting rate was observed indicates 
that scattered radiation was not contributing to the 
effect. Second, the source and detector might be 
polarized (because of permanent magnetization or 
by stray magnetic field) arid thus change the relative 
intensities and polarizations of the hyperfine lines. To 

Table I. Values of parameters used with theory of Sec. II 
to fit data in Figs. 6, 7, 8, and 9. 



Parameter 


Value 


Comment 


/ 


0.6 


Fraction of 14.4-kev gamma 
rays emitted without recoil. 


/' 


0.6 


Fraction of Fe" nuclei absorb- 
ing without recoil. 


a 


15 


Internal conversion coefficient. 


<r 


1.48XlO-»cm* 


Peak absorption cross section 
for no hyperfine splitting. 





2N<r f 


Thickness parameter of theory. 
(This value is twice the ex- 
pected value.) 


N 




Number of Fe" nuclei per cm' 
in absorber. (Obtained from 
weight of foil.) 




2345678 10 
ENERGY DIFFERENCE IN UNITS OF r 



Fig. 10. Transmission (relative to the vibrated condition) of 
enriched Fe" absorber obtained at \t and 4r after formation of 
the 14.4-kev state. The solid and dashed curves represent the 
theoretical predictions; the circles the experimental data points. 



test for static polarization of source and detector, 
transmission measurements were made with various 
orientations of the clamped source and absorber. No 
effect larger than the statistical accuracy of 1% was 
found. In another measurement, an upper limit of 5 
gauss was put on the local magnetic field, with a 
probable value close to the earth's field. These two 
observations make it unlikely that the source or 
absorbers were polarized. Third, because the measured 
linewidths were somewhat greater than they should have 
been, one should perhaps average over a range of energy 
shifts in the region of the nominal displacement AE. 
A series of curves were calculated by averaging the 
curves for a given AE over a Gaussian distribution of 
AE. Although the agreement between theory and 
experiment could be improved in some cases in this 
way, it was worse in other cases and no net improvement 
resulted. 

ACKNOWLEDGMENTS 

We are indebted to Dr. G. J. Perlow and Dr. S. S. 
Hanna for suggesting the original problem and for 
supplying the proper sources and absorbers for perform- 
ing the experiments. J. B. Baumgardner and A. Vander- 
gust built most of the special electronic circuits used. 



APPENDIX 



The integral 

1 r +- 
a'{t)= — I d<*- 
2iri J„ u 



■\i\ 



[2ibw I 
w 2 — wo' 2- twXJ 



(Al) 



can be evaluated by completing the contour on a 
semicircle in the upper half of the complex o> plane and 
finding the residues of the integrand. Since wo'^X, the 
exponent has singularities at w=±aj</+$tX. It is easily 
shown that the contribution from w=— wo'-f-ftA 
contains a factor X/W, so this term can be neglected. 



266 

520 LYNCH, HOLLAND, AND HAMERMESH 

We are then left with Substitution in Eq. (A4) yields 

J ~ giut 

2m J W -« -£*X **-— f Gdw 

Xexp[ib(co— W— 2^) _1 ]= — <p Gdw « / ib 



= — I <f Gdw+ <f GdoA. (A2) 



a>o— wo 
+«x™ ' J-o'+h-x J' V ' X(^)-i<«+ 1 >7 n+1 (26i/i) (A6) 



= -exp[W/-|Xf] £ ( ) 

<(ft*)-* (B+ 

[A] 

X(ft/)~ 



In the first integral we let z=w— wo— ?iX so that _. , , _ " 

= -exp[W<-§XOE 

1 y 1 n-1 LO) - 

Ji= — ® Gdw=-^— exp(W— 2^0 

2«.A*+ja 2« X(bt)- n ' 2 J n (2bW) (A6a) 



(A6b) 



= exppo> /— |X/+tft/(«o— wo')]- (A3) 



where S„ is the summand in Eq. (A6a). 
In the second integral we set z=w-a>o'-|«X so that We use the generating function (A5) in the first sum 

I j and find a term which cancels I h so we are left with 

Ii= — <p Gdco= — exp[io)o't— %\t] ., 

w/,r+ » 2iri «'(/)=«pCW/-ixo L [-— I 

K^x-i *— °° Lo) — 0>o'J 



dz r / ft\i »— »Lo)o-o) 

exp 



UzW-«o L V z/J X(bt)-»i*J n (2bW) 

1 « fi(wo _ o)o')T 

exp[W/-|X<] =exp[W/-§XO £ 

2m n=o L 6 J 



/£) <fe expU ( tz+- \ 1 (A4) 
«-0 (a) -0)o') n+1 L \ Z/J 



X(ft/)" /2 /„(2ftM). (A7) 



For large values of (o) — o) ') it is convenient to obtain 
From the formula for generating Bessel functions, a '(t) from the sum of (A3) and (A6a). Then we have 



exp[^(«-l/«)]= E «-/«(*), (A5) f r ft 

«— «> a (/) = exp[to)o /— jA/Jj — expj 

we find ' Lo)o— o)o' 



a'(/) = exp[zo)o'/— |X0J — expi h(o>o— o>o')2 

I LO)o— O)o' J 



exp[i(te+-)] = L i m (t/b)**z»J m (2bW). (ASa) + £ f — ) (bt)-"'V n (2bHi) \ (A3) 

L V Z/ J m— co «=1 \O) — 0)0 '/ 



267 



Effect of Radiofrequency Resonance on the Natural Line Form (*). 

M. N. Hack and M. Hamermesh 
Argonne National Laboratory - Argonne, III. 

(ricevuto I'll Ottobre 1960) 



Summary. — The form of Zeeman lines in the presence of a resonant 
rotating r.f . magnetic field is determined. The spontaneous emission distri- 
butions are obtained from the steady- state solutions, leading to a pre- 
diction of the splitting of Zeeman lines at rotation frequencies corre- 
sponding to well separated single- and multiple -quantum resonance fre- 
quencies, as well as at the resonance frequency in the Majorana case. 



1. - Introduction. 

Some time ago Bitter ( x ) and Pryce ( 2 ) indicated the possibility of a new 
method of detecting radiofrequency resonance, by means of certain changes 
in the radiation emitted by an atom when it is subjected to a radiofrequency 
field at a resonance frequency between its Zeeman levels in a constant mag- 
netic field. At that time the observation of such an effect appeared difficult 
to perform, since it would require large radiofrequency field amplitudes and 
high resolution of the emitted radiation in order to detect the changes pro- 
duced by applying the radiofrequency field. However, the possibility of an 
experiment on the nuclear Zeeman lines by means of the Mossbauer effect 
has renewed interest in this problem ( 3 ). 



(*) Work performed under the auspices of the IT. S. Atomic Energy Commission. 
f 1 ) F. Bitter: Phys. Mev., 76, 833 (1949). 

( 2 ) M. H. L. Pryce: Phys. Rev., 77, 136 (1950). We are indebted to Professor 
A. Abragam for bringing references ( x ) and ( 2 ) to our attention when we informed 
him of our results. 

( 3 ) A preliminary experimental attempt was reported by E. C. Avery, C. Little- 
john, G. J. Perlow and B. Smaller at the Allerton Park Conference, University of 
Illinois (June 1960). 



268 

4-9- **^.^.***^.*4-**4»9"?"9-*****4»9'********4"9»^ 44-4-4-4-4-4-4-J>4-4-4>4-4-4-4-4-4-4'4- 

2 EFFECT OF RADIOFREQUENCY RESONANCE ON THE NATURAL LINE FORM [547] 

In the present note we study the influence of radiofrequency resonance on 
the natural line form of the emitted radiation. The general theory is closely 
related to the steady-state solutions found in the study of multiple-quantum 
transitions ( 4 ). 



2. - The two-level ease. 

We first consider a system undergoing radiative decay from a pair of levels a 
and p to levels a' and p' respectively, in the presence of a rotating r.f . field 
which produces transitions between a and ft (Fig. 1). We wish to study the 
effect on the emitted spectral lines when the angular 

frequency of rotation co is close to the resonance f re- a r . 

quency o) a p= (E a — E p )lh between the upper states, q ( J 6 ^ 

and the amplitude of the rotating field is sufficiently 

large to produce an appreciable probability of trans- ' 

fer between a and ft during their mean life-time. We 

shall assume that both states decay at the same rate. a' — 

For the present we neglect the influence of other 
levels on the resonance a <-->/?. In the following * 



section the possibility of transitions to other levels Fig. 1. - Decay transi- 
will be taken into account, leading to corresponding tions in the presence of 
results also at the resonance frequencies associated a rf - field - 

with well separated single- and multiple-quantum 

transitions, as well as at the single resonance frequency in the case of equally 
spaced levels. 

To investigate the effect of the radiofrequency resonance on the natural 
line form, we study the solution of the system of equations 

(la) ib ao = H exp [i(co ap - co)t] b po + 2 s M \a'i l ex P C*K«' ~ ^xW 6 <*'i » 

x 

(16) ib a , h = B a . Hm exp [*(a> 1 — co^)t]b a0 , 

(lc) ib po = H exp [i(co — co^f] b ao + J H^,^ exp [i{co^, - co a )t] b^ , 

a 

(Id) ib fi% = H p , lg{fi0 exp [i(o a - topp) t] bp , 

subject to the condition that initially no photons are in the radiation field, 

(2) b a , h (0) = b fi , la (0) = . 



( 4 ) M. N. Hack: Phys. Bev. y 100, 975(A) (1955); 104, 84 (1956); H. Salwen: 
Phys. Rev., 99, 1274 (1955); C. Besset, J. Horowitz, A. Messiah and J. M. Winter: 
Journ. Phys. et Bad., 15, 251 (1954); J. M. Winter: Ann. Phys., 4, 745 (1959). 



269 



[548 J m. n. hack and m. hamermesh 3 

For H = 0, i.e., for zero amplitude of the radiofrequency field, these equa- 
tions reduce to the equations of Weisskopf and Wigner ( 6 ) for describing the 
radiative decay of excited states. For vanishing matrix elements of the inter- 
action with the radiation field, -H^^i = 0, etc., they would reduce to the 
equations describing the transitions produced by the radiofrequency field alone. 

In general eqs. (1) are too complicated to solve exactly. Nevertheless, for 
the cases of practical interest a satisfactory approximate solution can be ob- 
tained. For this purpose we look for solutions of the form 

(3a) b a0 = C a exv[—ipf], 

(36) b 0o = C p exp [— i(p + o> afi — a))t] . 

Substituting (3a) into (lb) and integrating, subject to the initial condition (2), 
and inserting the result 

GW + P — CO X 



and eqs. (3) into (la) gives 

(5) pC^HCp+C.^B 






1 — exp [i(w a<x > + p — co?.) t] 



(For simplicity photon occupation numbers have been suppressed.) 

It is now important to note that for the times of actual interest, i.e., during 
a long time interval which includes the mean lifetime of the decaying states, 
by virtue of the smallness of the decay constant (line width) compared to 
the line frequency, the last term in eq. (5) is practically constant ( 6 ). Denoting 



( 5 ) V. Weisskopf and *E. Wigner : Zeits. Phys., 63, 54 (1930). 

( 6 ) V. Weisskopf and E. Wigner: ref. ( 6 ). One can estimate the upper limit 
of the interval to be of the order of a few times r- 1 In (w /JT), for r < co . At the same 
time one obtains an estimate for the lower limit (non-linear decay contribution to the 
change in probability of remaining in the initial state at very early times) as being 
at most of the order of the period 2n/co . These results follow already in the Weisskopf- 
Wigner approximation, even with the usual neglect of the frequency variation of the 
matrix element and density of states. In this case the second term in the sum in 
eq. (5) leads to a factor 

OD 

r[exp[i(w-a> A -£;r)f| rln<1T 
^ rr^ dc °A = 2m + exp [£ rt]I , 



270 

4 EFFECT OF RADIO-FREQUENCY RESONANCE ON THE NATURAL LINE FORM [549] 

its value by — (ij2)r-C <x ( 7 ), we have 



(6a) 



In the same way we get 



v + -r\c a = nc,. 



(46) 
and 
(66) 



exp[— iico^' + p— co — co a )t] — 1 

Or' = tLR'fiKjfi - : , 



CO, 



p — CO — CO c 



HC a = [p + ^r+co^-co 



)<v, 



where by the restriction to the case of equal lifetimes r has the same value 
as in (6a). 

The simple eigenvalue problem, eqs. (6a, 6), has the solutions 



(7a) 
where 
(76) 
and 

(8a) 



1 i 

P± =2 (<» — (w fl/ j) + w±— -T, 



^ ± = ±K(co-co^) 2 + 4ff^ > 



± 






i± 



COtf — CO 



[(co - co„pY + 4H«]* 



(86) 



^../. 



1 

V2 



IT 



[(G)-0)^)« + 4fl-»]*] 



where r > 0, ( 7 ) and the last term furnishes the deviations. For all t>0 we have 
|I|<l/a>t, and for £ » 1/eo, I ~ {exp pia>£]}/*(co — ir/2)t. More refined estimates tairing 
into account the frequency variation of the matrix element and density of states lead 
to similar results. The dependence of the decay constant on the r.f. quantities in our 
case (co — co a/3 , H ) is also weak since these are very small, of the order of the line 
width, compared to the frequency of the line. Decay theory has been discussed from 
a different viewpoint, based on the characteristic functions of the total-energy operator, 
by L. A. Khalfin: Sov. Phys. Journ. Exp. Theor. Phys., 6, 1053 (1958). 

( 7 ) We suppose the imaginary parts of the decay constants to be absorbed into 
the energies E a and Ep. Then Tis purely real and moreover, as one readily verifies, >0. 
We take r strictly positive, since we naturally are not interested in oases where selection 
rules forbid the decay. 






271 



[550] 



M. N. HACK and M. HAMERMESH 



The particular superposition satisfying the initial condition that the system 
is in the state a0 at time t = is given by 



(9) 



\o = Q \« ex P [- W+ Q + &-* ex P [- *P- fl » 

. ft /J0 = ( C +« +/> eX P [— *+fl ■+■ -« C -/3 eX P [— V- Q) eX P t— *( 



W 



c«)q . 



The probabilities of finding the system subsequently, at time t, in the states 
a0 or /?0 are therefore 



(10) 

where 

(11) 



\b ao (t)\*=(l-P)exj>[-n], 

\b po (t)\*=Pex V [-rt], 



42? 2 1 

(a,-«, tf ). + 4H i Sin2 2 «» ~ »^" + *** * ' 



i.e., the Eabi transition probabilities multiplied by the exponential decay 
factor exp[— TV], 

The probability amplitudes for finding the system in the lower levels after 
decay, with a photon of frequency a> x or a> a respectively in the radiation field, 
are ( 7 ) 



(12) 



Moo) = H a > a lr}+C +l 
bp\oo) =HpJr)+C +t 



0>X — CD aa > — p H 

1 

<*>o — 0>*/3' — P+ + CD 



+ rj-C- x 



(Dl—O) «• 



+ V-C. 



0) a — 0) a 



P- + 0) 



where the constants r] ± (\r]+\ 2 4- \v~\ 2 = 1) determine the particular superposi- 
tion satisfying an arbitrary initial condition. 

For the initial condition corresponding to (9), rj+=C +i(X and r}-=G_ % , 
For 

(13) o) = w a p (r.f. resonance), 

the corresponding emission probabilities are therefore 

(a> A -a)o) 2 + r*/4 



(14) 

where 
(15) 



b^ a -(oo) 



( #«*'| 2 



b^(oo)\^ = \m^Y 



[((o x - <wo) a - r*/4 - # 2 ] 2 + r*(m -o) y' 



[{co a - o>i) 2 - r^/4 - b" 2 ] 2 + r 2 (o> - o)o)z 



0) n = O), 



272 



6 



EFFECT OF RADIOFREQUENCY RESONANCE ON THE NATURAE LINE FORM 



[551] 



Similarly for the condition that the system is initially in the state /?0 at time 
*=0, 



(16) 






m 



[{o) X - co ) 2 - r*/4 - H*Y + r*(a> x - a> ) 2 ' 

(ayo-cooy + r'l* 

[{co a - co' y - T^/4 - F 2 ] 2 + r*{a> a - co ) 2 ' 



The effects of the r.f . field on the emitted spectral lines are readily seen in 
the solutions (14) and (16). They persist and take an even simpler form in the 
case of equal populations (and random phase difference) in the levels a and /?. 
In this case the emission distribution is the superposition with equal weights 
of the distributions for the pure steady-state solutions where the components 
are displaced without change of shape. 

Averaging over the inital states and summing over all the final photon 
states at the energy tiw K , we obtain for the line a -> a' the photon frequency 
distribution 

(a>x - o) ) 2 + # 2 + T 2 /4 

w ) 2 * 



(17) 



P(CO A )res = — 



4tt [{(o x - co ) 2 - T 2 /4 - # 2 ] 2 + r*((o x 



The similar distribution for the line f$ ->/?' is obtained by replacing co by co . 
The corresponding results in general, i.e., not restricted to r.f. resonance, are 



(18) P(mx) = 



a>x—a) 



±71 



T + T + I {( ° ~ W ^ )2+ Hi -(°>- <°«J (^A-«0- ™** \ 



<o x — 0) Q 



0> — (O*8 



\ 2 1 3 1 

)____ (co _^ )2 _^ 2 



+ r«u A -to - 



co — (O xfi 



and the similar relation for the line (i -> ft' obtained by replacing co by 
a) a5 , — co and changing the sign of the last term in the numerator. 

Fig. 2 shows the behavior of the emission distribution at resonance, for 
increasing values of the radiofrequency field amplitude. When the field ampli- 
tude is sufficiently large that the angular frequency of oscillation between a 
and /? is comparable to the decay rate, the spectral lines a -> a' and ft -> /?' 
each exhibit two peaks. The distribution P(a> A ) re , is the superposition of two 
Lorentzian ( 8 ) components of equal intensity, and widths equal to r, centered 
at ft> ± H . 



( 8 ) The frequency dependence of the factor (\E aet - \ t e(a> A )&Q leads to asymmetry 
of the normal line ( 5 ). For the frequencies of interest, however, because of r<eo , 
we make a negligible error by replacing this factor for simplicity by its value at the 



273 



[552J 



M. N. HACK and BI. HAMERMESH 



One verifies further that the splitting of the spectral lines appears only 
for co close to resonance. As the relative deviation of co from resonance in- 
creases, both lines tend to their natural forms (the intensity of the off-frequency 
component drops to zero). The emission distribution near resonance in 
the two-level case is illustrated in Fig. 3. 



(1/2) 




P M2 ( "*ks 



H = 2T 



CO- 



CO =U) r 




Ct)-Ct) r = H 



Fig. 2. - Emission distribution P^co^) 
at resonance for increasing r.f. ampli- 
tude. 



Fig. 3. - Emission distribution P^co^) 

for fixed R=\r and varying deviation 

from resonance. 



line center. At large w A it tends to zero by virtue of the retardation, as required for 
convergence of the total radiated energy. For small co A (frequencies up to and even 
beyond the line frequency) it has the approximate value jTco^/a) . 



274 



EFFECT OF RADIO-FREQUENCY RESONANCE ON THE NATURAL LINE FORM [553] 



3. - Arbitrary number of levels. 

The preceding treatment of the. decay from two levels can readily be ex- 
tended to the case of an arbitrary number of decaying states ( 9 ). We will 
show that the splitting of spectral lines by r.f . fields also occurs at distinct mul- 
tiple-quantum resonance frequencies, as well as at the resonance frequency 
in the case of equally spaced levels. 

For this purpose we consider a rotationally invariant system which inter- 
acts, through magnetic moments yJt and y a / a , with an external magnetic 
field consisting of a constant field 3^ Q and a perpendicular rotating field of 
constant amplitude Jf and angular velocity co, which causes transitions be- 
tween the levels of the system in the constant field. 

Omitting for the moment the interaction with the radiation field, and 
choosing the z direction along the constant field and the x axis along the 
direction of the rotating field at time t = 0, we have for the Hamiltonian of 
the system 

(19) 8=H + H', 
where 

(20) H'= {y x J x + yzJ^'W? cos cot)i + (^ sinco«)y] . 

H contains the interaction with the constant field and the J, -J 2 interaction, 
and commutes with the z -component of the total angular momentum 

The Schrodinger equation takes on a more convenient form if we transform 
to the co-ordinate system of the rotating magnetic field ( 10 ) by means of the 
unitary operator U= exp[iJ s a>*/^]. Then 

(21) <p = Uy> = exTp[iJ z wtlfi]y) 
satisfies 

(22) ih^ = W<p, 



( 9 ) When the calculation is extended in addition to an arbitrary number of lower 
levels to which a given state can decay, the decay constants combine additively to 
give the total width ( 5 ). It is this summation which leads in practice to the (approx- 
imate) equality of lifetimes of the decaying states. When several states decay to the 
same level, cross terms appear which are rapidly oscillating for co > J*. 

( 10 ) I. I. Rabi, N. F. Ramsey and J. Schwinger: Bev. Mod. Phys., 26, 167 (1954). 



275 



[554] m. n. hack and m. hamermesh 9 

where the transformed Hamiltonian 

(23) W=H -coJ z +V 
with 

(24) V= y x Jv>X+y % J*# 

is independent of the time. 

The transformed Schrodinger equation' (22) has steady-state solutions 

(25) cp = w-exp [— iwt] . 

By means of this transformation, the problem is reduced to the solution 
of an eigenvalue problem. As in the two -level case treated in the previous 
section, the introduction of the decay transitions effectively changes w accord- 
ing to 

(26) w -+ w -2r, 

where r is the decay constant. Here, as before, we consider the case where 
all levels decay at the same rate. In this case the eigenvalue problem is the 
same as in the absence of decay. (In the general case, the effect of decay is 
to introduce constants }i/\, HT 2 > ••• into the diagonal terms of the eigen- 
value equations. In the case treated here, all the .T's are equal). 

For well separated resonance frequencies and for a> in the neighborhood 
of a resonance frequency, it has been shown ( 4 ) that the eigenvalue problem 
can be reduced to a two-level problem, even in the case of the higher order 
multiple-quantum transitions. 

Equations (1) then apply directly, with H replaced by an expression of 
the type of higher order perturbation theory, ( n ) and o> — co^ replaced by 
a quantity Q' ( 12 ). The splitting of the spectral lines thus occurs for each of 
the resonance frequencies, including the ones corresponding to multiple- quan- 
tum transitions. Only the two lines emanating from the particular initial and 
final levels of the given resonance transition are split by the application of 
the r.f . field. This follows from the fact that only these levels have appreciable 
components in the corresponding steady-state solutions which become almost 
degenerate at the given resonance. 

The opposite case of equally spaced levels, where several resonance fre- 
quencies coincide, can be treated by similar methods. Apart from the field- 



( u ) M. N. Hack: reference ( 4 ), eq. (51) for 8. 
( 12 ) M. N. Hack: reference ( 4 ), eq. (42), (50), (51). 



276 



10 EFFECT OF RADIOFREQUENCY RESONANCE ON THE NATURAL LINE FORM [555] 

independent term of w, the steady-state solutions for this case (Majorana case) 
can be expressed in the form ( 13 ) 

(27) O£=d£.(0), 

(28) wf = ju[{oj — w r Y + 4# 2 ]* , 
where 



{29a) cosO = 

{29b) sin d = 



[{a) — (o r y + 4# 2 ]* 

2H 

[(co — a) r y + 4B" 2 ]* ' 



co r = yjJ^ , 2H = y j yf 1 and the d ( ^ m are representations of the rotation group. 
Applied to the decay problem, they lead to the probability amplitudes 

(30) V» m = df m exp {- i(m[co - co r ] + (x[{m - co r ) 2 + 4# 2 ]* - \ir)t) , 

(31) &; ( £(oo) = df m H t 



coa - K + m[w - oj r ] + j*[(a> - co r Y + 4fl- 2 ]*) + JiT ' 
In the case of equal populations and random phases, this gives 

(32) p;>,) = ^7i£2I<U0)I 2 - 

1 

*{co7- (co + m[(o - oj r ] + /x[{a7^1oJ*~+lH>¥)y + T 2 /4 ' 
i.e., a superposition of normal components with widths r, centers at 

co +7n[co — to r ]+ix[{w — co r ) 2 4-4ff 2 ]^, ju = — j, — 7+1, ...?, 

and intensities proportional to the squares of the coefficients of the rotation 
matrices. 

In this case, each line thus splits in general into a number of components 
equal to the multiplicity 2j-\-l. However, in certain cases fewer components 
may appear, because some of the intensity coefficients vanish. At resonance 
{(o = co r ), this occurs when the multiplicity is odd, for the central level (the 
next to last and alternate components are absent) and besides the central 



( 13 ) H. Sal wen: reference ( 4 ), eq. (38), (39). 



277 



[556] 



M. N. HACK and M. HAMERMESH 



11 



level for all the alternate levels starting from the next to last (the central 
component is absent). 



- 


ct)-(t) r *UH r\ 


P ( 3 3 / 2 2 W 




- 


0)-OJr --2H n \ \ 




H-f 


- 


0)-W r -H //1Y\%" 






- 


0J=CJ r //// 






"* 


1 1 1 1 1 


1 1 1 


1 1 I 



0J f 



*V3y 



a) 



Photon frequency t co^ 




Photon frequency , co* 

Fig. 4. - Emission distribution P^ioy^) for fixed H=^T and varying co>(o r . (a) ra=f ; 
(6) m=\. The corresponding curves for co<co r are obtained by reflection in the ver- 
tical axis through co . The curves for negative m are obtained by reflection in the 
same axis and displacement to the corresponding mean emission frequency. 



In the particular case of j = \, eq. (32) reduces to eq. (18) ff. (Figs. 2 
and 3). The graphs of P^ico^) for the case of j = f are shown in Fig. 4a and b 
for fixed H = T/2. 



278 

12 EFFECT OF RADIOFREQUENCY RESONANCE ON THE NATURAL LINE FORM [557) 



RIASSUNTO (•) 



Si determina la forma delle linee di Zeeman in presenza di un campo magnetico 
rotante risonante a r.f. Le distribuzioni della emissions spontanea vengono ottenute 
dalle soluzioni per lo stato continuo, e portano alia predizione della scissione delle linee 
di Zeeman sia per frequenze di rotazione pari alle ben distinte frequenze di riso- 
nanza quantistiche singole e multiple, sia per la frequenza di risonanza nel caso 
di Majorana. 



(*) Traduzione a cur a della Redazione. 



279 



Volume 4, Number 8 



PHYSICAL REVIEW LETTERS 



April 15, I960 



RECOILLESS RAYLEIGH SCATTERING IN SOLIDS 

C. Tzara and R. Barloutaud 

Centre d' Etudes Nucleai res de Saclay, Gif-sur-Yvette (Seine-et-Oise), France 

(Received March 21, 1960) 



Using the Mossbauer effect, photon sources 
and analyzers extremely selective in energy are 
now available. We study here with such an ana- 
lyzer the recoilless Rayleigh scattering by atoms 
in solids. 

This effect is related to the x-ray diffraction by 
crystals as follows. The interference at the ex- 
act Bragg angles occurs when the scattering is 
elastic with respect to the lattice as a whole, 
that is, without any phonon exchange. Debye and 
Waller have calculated the reduction in intensity 
of x rays scattered at the Bragg angles in a 
solid at temperature T, 1 

.x 



exp 



2 k6 L 4 x •'o e U - J' 



(1) 



where x = T/Q, d is the Debye temperature, and 
E R = (E 2 /Mc 2 ) (1 - cosfl) is the recoil energy 
given to the free atom by a photon of energy E 
scattered at the angle Q. 



In the present work, where we detect the elas- 
tic scattering directly by an energy selection 
instead of analyzing a diffraction pattern, the 
factor cpT is the relative number of photons scat- 
tered without energy change. It is clearly the 
same factor which gives the proportion of re- 
coilless y rays in the Mossbauer effect 2 ; in that 
czseE R =E 2 /2Mc 2 in Eq. (1). 

In order to measure the factor <p T , we have 
studied the Rayleigh scattering for several ma- 
terials: Pt, Al, graphite, and paraffin. The 
23.8-kev photons emitted by Sn 119 * were scat- 
tered at 50° ± 5° and absorbed by a Sn 119 foil 
40 mg cm" 2 thick (almos* completely black for 
the recoilless photons 3 ) (Fig. 1). The scatterers' 
thicknesses were such that the transmission of 
the y rays was of the order of 10%. 

The Rayleigh- scattered photons are accom- 
panied by inelastically scattered photons (Raman, 
Compton), considerably shifted in energy, so 
that the selective abs'orption in Sn 119 occurs only 



405 



280 

Volume 4, Number 8 PHYSICAL REVIEW LETTERS April 15, 1960 




Table I. Experimental and calculated values oi<pj 



Wet 



Pt 


1.05 


Al 


2.15 


C 


4.13 


CH, 


5.30 



calc 



0.27 ±0.03 0.72 ±0.09 0.80 

0,19 ±0.016 0.92±0.09 0.62 

0.10 ±0.01 0.79±0,09 0.68 

0.020 ±0.01 



FIG. 1. (1) Sn 119 * source; (2) scatterer; (3) bismuth 
stopper; (4) 40 mg cm" 2 Sn m foil (71.5% Sn 119 ); (5) 62 
mg cm" 2 Pd foil absorbing Sn x rays; (6) 1.5 mm 
Nal(Tl) scintillator and photomultiplier. 



for a fraction a of all the scattering processes; 
a is extracted from the form factors given by 
Compton and Allison. 4 

We have measured the relative decrease A of 
counting rate between room temperature T 1 - 300°K 
and T„ = 80°K. The recoilless scattering proportion 



T 2 



I A ' € 

a f 2 -e 



where / 2 is the ratio of recoilless emission of the 
Sn 119 * source and e is its self- absorption at T 2 . 
Here / 2 = 0.32 ±0.01 5 and e = 0.05±0.01. We neg- 
lect the small recoilless emission at 300°K which 
introduces a negligible correction for (p T . 

The results are given in Table I. The agree- 
ment between the calculated and experimental 
values of <p is reasonably good, especially when 



we notice that the Debye temperatures are de- 
duced from specific heat measurements rather 
than from x-ray diffraction. 

We have also, using a thin Sn 119 foil as a scat- 
terer, observed at low temperature the resonant 
Mossbauer scattering. 5 

This method extends the range of solids which 
can be studied by means of the Mossbauer effect 
or by x-ray diffraction. 

It is a pleasure to acknowledge interesting dis- 
cussions with Dr. Abragam, Dr. Cotton, and 
Dr. Jacrot. 



1 I. Waller, Ann. Physik 79, 261 (1926). 

2 R. L. Mossbauer, Z. Physik 151, 124(1958). 

3 R. Barloutaud, E. Cotton, J. L. Picou, and J. 
Quidort, Compt. rend. 250, 319(1960). 

*A. K. Compton and S. K. Allison, X-Rays in Theory 
and Experiment (D. Van Nostrand Company, Inc., 
Princeton, New Jersey, 1947), 2nd ed. , p. 781. 

5 C. Tzara and R. Barloutaud, Compt, rend, (to be 
published). 



406 



281 



PHYSIQUE NUCLEAIRE. — Sur la possibility de mettre en evidence la coherence 
de phase dans la diffusion de resonance des rayons y par des noyaux ato- 
miques. Note de M. Alfred Kastler, presentee par M. Gustave Ribaud. 



Lorsque la diffusion de resonance des rayons y par des noyaux atomiques d'un 
rgseau se fait sans recul du noyau, la coherence des radiations diffusees doit se 
manifester par une repartition d'intensite analogue a celle de la diffraction des 
rayons X par les electrons d'un cristal. 

La decouverte faite par Mossbauer ( d ) que des rayons y de faible energie 
peuvent etre emis ou absorbes par des noyaux situes dans un reseau 
cristallin sans perte d'energie due au recul du noyau, a souleve un interet 
considerable. La finesse extraordinaire des rayons y obtenus dans ces 
conditions permet d'aborder des investigations nouvelles : L'etude de la 
largeur de raie par analyse cinetique grace a l'effet Doppler, l'etude par 
ce meme procede de la structure Zeeman et de la structure hyperfine 
de la raie, la mise en evidence par une experience de laboratoire de l'effet 
Einstein, c'est-a-dire de la difference de frequence entre deux etalons 
identiques liee a la difference de potentiel gravifique ( 2 ). D'autres etudes, 
confinees jusqu'a present au domaine optique vont pouvoir etre etendues 
au domaine des rayons y. Citons a titre d'exemples les mesures de varia- 
tion de frequence dans la diffusion des ondes electromagnetiques par les 
ondes d'agitation thermique (effet Brillouin), les etudes des effets de pola- 
risation et d'anisotropie spatiale des radiations de resonance lorsque 
emetteur et absorbant sont soumis a des champs magnetiques de directions 
variees ( 3 ), enfin les applications des methodes optiques de spectroscopic 
des radiofrequences ( 4 ). 

Une autre question interessante va pouvoir recevoir maintenant une 
reponse experimentale, celle de savoir si les radiations de resonance electro- 
magnetiques diffusees par des centres resonnants presentent des effets de 
coherence de phase. Une telle coherence a ete explicitement admise par 
Weisskopf ( 5 ). Si la radiation de resonance diffusee est totalement inco- 
herente, sa repartition spatiale doit etre continue et conforme aux rela- 
tions de correlation angulaire qui ne font entrer en jeu que le caractere 
multipolaire de la transition spectrale et les nombres de spin des niveaux 
qui bordent la transition. Si, au contraire, la radiation diffusee est partiel- 
lement coherente, comme on peut le supposer pour la composante Zeeman 
qui ramene le noyau au niveau magnetique initial, il faut s'attendre a 
observer des effets d'interferenCe. Une diffraction selective doit se faire 
dans les directions de von Laue-Bragg, ou dans celles des anneaux de 
Debye-Scherrer, suivant que la matiere diffusante est mono- ou poly- 
cristalline. 



282 



( * ) 

Dans le domaine optique, lorsqu'il y a diffusion resonnante de la lumiere 
par les atomes d'une vapeur sous faible densite, la lumiere diffusee ne 
presente pas de caracteres de coherence. Dans ce cas, l'irregularite des 
positions des centres diffusants et les fluctuations de densite masquent 
completement la coherence (si elle existe) de Facte de diffusion elementaire 
(tout comme l'irregularite des mouvements de translation de ces centres 
masque la finesse « naturelle » de la raie). L'apparition, aux fortes densites 
de vapeur, d'une reflexion reguliere sur la face d'entree de la vapeur ( 6 ) 
revele la coherence. 

Dans le cas de la diffusion resonnante de rayons y par des noyaux, 
le caractere quantique du phenomene est predominant. Ce caractere 
n'exclut pas les proprietes de coherence. Dans le cas de la diffusion de 
photons y par les noyaux d'un reseau, le noyau diffusant devient identi- 
fiable lorsqu'il recule (par exemple, il peut quitter un noeud du reseau 
pour se mettre en position interstitielle). Dans ce cas, la faculte d'inter- 
ference du photon diffuse se trouve detruite. Mais lorsqu'il n'y a pas de 
recul nucleaire, lorsque c'est le reseau tout entier qui encaisse la quantite 
de mouvement du recul, le noyau diffusant n'est pas identifiable. Dans ce 
cas, la coherence de phase du rayonnement liee a l'arrangement spatial 
periodique des centres diffusants doit se manifester. 

II faut remarquer que la coherence n'est que difficilement observable 
lorsque les noyaux diffusants sont ceux d'un isotope irregulierement 
repartis dans un melange isotopique. L'observation de la coherence par- 
faite necessite un arrangement spatial regulier des noyaux diffusants, done 
Femploi d'un cristal forme d'un isotope pur ou fortement concentre. 
Lorsqu'on ajoute a cet isotope actif des proportions croissantes d'isotopes 
inactifs dans le reseau, l'intensite des raies de diffraction diminue au profit 
du fond continu incoherent et le contraste se trouve affaibli. 

II faut pouvoir distinguer la diffusion de resonance nucleaire coherente 
de la diffusion normale, egalement coherente, par les electrons des atomes 
du reseau. Dans le cas de 57 Fe, ce dernier phenomene est d'ailleurs d'inten- 
site negligeable ( 7 ). Les deux effets sont separables en comparant l'in- 
tensite diffusee avec une source immobile et une source mobile. Le mou- 
vement de la source permet de supprimer la resonance nucleaire, il ne 
modifie pas l'intensite de la diffusion des rayons y par les electrons. II est 
a prevoir que la diffraction nucleaire donne des raies de diffraction beau- 
coup plus fines que la diffraction electronique a cause de la grande finesse 
spectrale des radiations de resonance nucleaire. La localisation precise des 
noyaux dans le reseau donne lieu a des caracteres particuliers : les franges 
d'interference d'ordres eleves doivent etre intenses. 

II faut noter qu'un cristal emetteur de substance-fille ( 57 Fe) conte- 
nant des noyaux emetteurs de substance-mere ( 57 Co) donne lieu au pheno- 
mene d'autodiffusion dont la coherence peut se manifester par des lignes 
de Kossel ( 8 ). 



283 



( 3 ) 

(*) R. J. Mossbauer, Z. Physik, 151, 1958, p. 124; Naturwissenschaften, 45, 1958, 
p. 538; Z. Nalurforschung, 14 a, 1959, p. 538. 

( 2 ) Craig, Dash, Mc Guire, Nagle et Reiswig, Phys. Rev. Lett., 3, 1959, p. 221; 
Lee, Meyer-Schutzmeister, Schiffer et Vincent, Phys. Rev. Lett., 3, 1959, p. 223; 
R. V. Pound et G. A. Rebka, Phys. Rev. Lett., 3, 1959, p. 43g et 554; J- P- Schiffer 
et W. Marshall, Physik. Rev. Lett, 3, 1959, p. 556. 

( 3 ) A. C. Mitchell et M. W. Zemansky, Resonance Radiation and Excited Atoms, 
Cambridge, University Press, 1934; P. Pringsheim, Fluorescence and Phosphorescence, 
Interscience, 1949. 

(*) A. Kastler, Nuovo Cimento, 6, 1957, Supplemento n° 3, p. 11 48. 

( 8 ) V. Weisskopf, Ann. Phys., 9, 1939, p. 23 (voir particulierement p. 25-26). 

( 6 ) R. W. Wood, Physical Optics, Mac Millan, New- York, 3* ed., 1934, p. 534; J. L. 
Cojan, These, Paris, 1953, Ann. Phys., 9, 1954, p. 385. 

( 7 ) S. S. Hanna et coll., Phys. Rev. Lett, 4, i960, p. 28. 

( 8 ) W. Kossel et H. Voges, Ann. Physik, 23, 1935, p. 677. 

(Laboratoire de Physique de I'Ecole Normale Superieure, 
24, rue Lhomond, Paris, 5 e .) 



284 



Resonant Scattering of the 14-keV. lron-57 

y-Ray, and its Interference with Rayleigh 

Scattering 

When radiation is scattered by a resonator, the 
question of its identity of frequency and coherence of 
phase with the incident radiation is not simply 
answered except for an infinitely narrow incident 
line, when the scattering is fully coherent (see, for 
example, the discussion by Heitler 1 , written in the 
context of atomic resonances but equally relevant to 
nuclear ones). 

Coherence may be experimentally proved by 
observing interference between the resonance radiation 
and some other form of scattering that is known to be 
coherent ; for y-rays, one would naturally look for 
interference with the Rayleigh (elastic electronic) 
scattering. In the long-studied 411 keV. resonance of 
mercury- 198, such interference is unobservable 2 , 
because of the thermal broadening of the line as 
well as for other reasons. It might just be observable 
in samarium- 152, where the natural width of the 
961-keV. line is not completely negligible 3 , 4 in com- 
parison with the thermal ; but the best conditions 
appear to be provided by the very narrow low- 
energy lines discovered by Mossbauer 5 in which a 
substantial part of the radiation is unaffected by 
recoil or thermal broadening. 

After preliminary experiments in collaboration 
with Dr. B. S. Sood, in which the resonant scattering 
in metallic iron of the 14-keV. line of iron-57 was 
found to have the expected intensity, we decided 
that a foil of 65 per cent iron-57, electroplated on 
thin copper by the Isotope Division, Atomic Energy 
Research Establishment, Harwell, would give com- 
parable amplitudes of resonant and Rayleigh scatter- 
ing at convenient angles. We in fact chose an angle 
of 43 ± 2°, which would include the (211) Bragg 
reflexion. The geometry of the apparatus is indicated 
in Fig. 1 ; the source 6 , about 20 mc. of cobalt-57 in a 
matrix of iron- 56, was mounted on a 30 c.p.s. vibrator 
so that the exact energy of the emitted radiation 
could be oscillated through the resonant energy. 
Auxiliary apparatus enabled the intensity of trans- 
mission through, or scattering from, the foil to be 
plotted automatically as a function of speed, the 
abscissae of the resultant graphs extending over 



285 



*&^^&$ 



>» ~* «. 








IV 



5 cm. 



Fig. 1. Diagram of experimental arrangement. B, vibrator; 
S, source ; C, proportional counter ; F, iron foil. The shaded 
blocks are heavy alloy shielding. The different positions of the 
source and vibrator are : I, for scattering ; II, for transmission ; 
III and IV, for checking absence of shift in transmission 



12,000 



11,000 - 



3 10,000 

» 3,000 



2,500 



2,000 



cpoo^ 



cP°° 



o°o 



-2 +2 +6 

4 +4 



+ 81 + 8 



+ 6 +2 
+ 4 



Velocity of source (10 -2 cm./sec.) 

Fig. 2. Plots of intensity versus velocity for transmission (dots) 
and scattering (circles). The zero of the velocity scale is fixed 
only by the position of the transmission minima. Positive 
velocities are for the source approaching the foil. The position 
M on the velocity scale is the point at which the acceleration 
reverses 



rather more than half a cycle of vibration, so that 
passages through zero velocity in both senses of 
acceleration were displayed. 

Fig. 2 shows the transmission minima and scattering 
maxima obtained ; it will be seen that the scattering 
peaks are somewhat broader and slightly nearer to 
one another than are the absorption minima. 

The reduced separation of the peaks was, however, 
due entirely to the shift of one of them ; presumably 
the whole pattern had suffered an instrumental drift 



286 

26,000 



23,000 



» '20,000 
| 8,000 

O 
O 



7,000 



6,000 



0^0 



000° 



^oo%*P*W 



j_L 



+4+2 0-2-4-6 -8 M-S -6-4-2 0+2+4 + 6 
Velocity of source (10~ 2 cm./sec.) 

Fig. 3. Plots of intensity versus velocity for a separate experiment 

in which, as indicated on the velocity scale, negative velocities 

are displayed in the centre 



during the long periods of operation that were 
necessary. We repeated the measurements with the 
apparatus set to display negative, instead of positive, 
speeds at the centre, alternating transmission and 
scattering measurements and adding the results of all 
sets of each kind. Fig. 3 shows that the peaks now 
appeared, as expected, more widely spaced than the 
dips, with no evidence of general drift. Thus, assuming 
the transmission dips to occur at exactly zero velocity, 
we can say that the maximum scattering occurs when 
the source is approaching the scatterer with a velocity 
corresponding to a fraction of the line-width. 

It remains to verify that the dips correspond 
precisely to zero velocity ; this need not be so if 
source and scatterer materials are not identical. This 
point was checked at the suggestion of Mr. D. A. 
O'Connor, by successive transmission experiments 
with the source assembly set in the positions marked 
III and IV in Fig. 1 ; this amounts to reversing the 
velocity-scale about the true zero. No change in the 
dip position was found. 

If there is some coherence of phase between 
resonant and Rayleigh scattering, we should expect 
them to be in quadrature at exact resonance, moving 
towards coincidence of phase on the high-frequency 
side. Thus, on top of the constant intensity of the 
Rayleigh and the sharply peaked contour of the 
resonant scattering, there should be added a dis- 
persion-type curve representing the interference 
between them, with its maximum a fraction of a 
line-width to the positive side of zero velocity. The 



287 



observed shift and broadening are probably to be 
explained in terms of such coherence, but the measure- 
ments are not yet detailed enough to show clearly 
the asymmetry of line-shape that is also to be expected 
or to determine what percentage of coherence exists ; 
full coherence is not to be expected when the incident 
line is of similar breadth to that of the resonance. 

We have also observed the resonant scattering 
from ordinary iron (2-2 per cent iron-57) at a mean 
angle of 90°, in the presence of a relatively large 
intensity of Rayleigh scattering. We found neither 
shift nor broadening of the scattering resonance, 
which is consistent with the expectation that, at this 
angle of scattering in a magnetic dipole transition, 
the resonant component will be polarized at right 
angles to the Rayleigh. 

It is to be expected that the narrow -line resonance 
radiation will show, in its scattering from crystals 
containing iron-57, interference phenomena similar to 
those of X-rays. Interesting differences may arise 
from the resonant absorption which accompanies the 
scattering, from the narrowness of the line as com- 
pared with X-ray lines, and from the simple way in 
which the exact energy, and with it the phase of the 
resonant scattering, may be varied. 

P. J. Black 
P. B. Moon 

Department of Physics, 
University of Birmingham. 

1 Heitler, W., "The Quantum Theory of Radiation", chap. 3 (Oxford 

Univ. Press, 1944). 
* Moon, P. B., Proc. Phys. Soc, A, 63, 1189 (1950). 
8 Grodzins, L., Phys. Rev., 109, 1014 (1958). 
4 Moon, P. B., and Sood, B. S., Proc. Roy. Soc, A, 257, 44 (1960). 
•Mossbauer, Pv. L., Z. Phys., 151, 124 (1958); Naturwiss., 45, 538 

(1958). 
•Chackett, G. A., Chackett, K. F., and Singh, B., J. Inorg. Nucl. 

Chem., 14, 138 (1960). 



288 



The Mbssbauer Effect in Tin from 120 °k to the Melting Point 

By A. J. F. BOYLE, D. St. P. BUNBURY, C. EDWARDS and 

H. E. HALL 

The Physical Laboratories, University of Manchester 
Communicated by B. H. Flowers; MS. received \%th June 1960 

Abstract. Measurements have been made of the intensity of the recoilless 
resonance absorption of the 24 kev y-ray from the decay of 119 Sn m in metallic 
tin from 120°k to the melting point. Values of the Debye-Waller factor deduced 
from these results tend towards the values calculated for a Debye of 142 °k 
at low temperatures; the behaviour of the Debye-Waller factor at higher 
temperatures indicates considerable anharmonicity of the lattice vibrations. 
Comparison with evidence from the thermal expansion and specific heat suggests 
that the quartic term in the interatomic potential is positive, and that the ratio of 
quartic to cubic terms is of the same order as the ratio of cubic to quadratic terms. 

In the last few degrees below the melting point the resonance absorption shows 
a rapid drop accompanied by an increase in line width. It is suggested that this 
effect is due to enhanced self-diffusion in the solid, and it is estimated that the 
diffusion coefficient reaches a value of 10 -8 cm 2 sec -1 about 0-6°k below the 
melting point. 



§ 1. Introduction 

Th e primary purpose of the experiments to be described in this paper was 
to investigate the way in which the recoilless y emission discovered by 
Mossbauer (1958) is affected by the transition from solid to liquid; for 
this purpose the 24 kev y-ray of 119 Sn m in metallic tin was used. The experiments 
show that the effect disappears continuously in the last few degrees below the 
melting point, and this result is attributed to self- diffusion in the solid. In 
addition, measurements of the intensity of the effect down to liquid air tempera- 
tures have yielded some information about the nature of the atomic vibrations in 
tin. 

For the purpose of this paper it is convenient to express the emission of y-rays 
from a solid in a way analogous to the elegant result derived by Van Hove (1954) 
for the scattering of x-rays or neutrons by an assembly of atoms. Van Hove 
shows that the differential cross section for scattering with momentum change 
fix, and energy change ha* is proportional to a quantity »S(x, a>) which is the Fourier 
transform in space and time of a time dependent pair correlation function. 
S(x, a>) is given by 

S(x,a>)=J-r e^/Bexpt-m.r^jexptm.r^)]), (1) 

^ 7r J — oo ij 

where r t (t) is the Heisenberg position operator of the ith atom and the symbol ( ) 



289 



130 A. J. F. Boyle, D. St. P. Bunbury, C. Edwards and H. E. Hall 

denotes an average value in thermal equilibrium. For emission of a y-ray from 
a given atom the equivalent result (Marshall and Schiffer, private communication) 
is that the probability of the emitted y-ray having wave number k is 

P( k )= o- P ex P [ic{k-K)t] exp (- £r y |*|)<exp[-xk. r(t)] exp [ik. r(0)]></f, 

^J -00 

(2) 

where hck is the energy of the y transition and F y its width. The correlations 
in (e"-* k - r(/) e* k - r(0) ) at small times give the emission of y-rays with recoil; at large 
times this factor tends asymptotically to 

exp(-2W0 = exp(-*V), (3) 

where x 2 is the mean square displacement of the atom (assumed Gaussian) in the 
direction of emission of the y-ray. The factor (3) is the Debye-Waller factor, 
familiar in x-ray crystallography, and from Eqn (2) we see that this fraction 
of the y-rays is emitted as a sharp line with the energy and width of the nuclear 
transition. The cross section for resonance absorption is likewise multiplied 
by this factor. For a Debye solid 

where E y is the energy of the y-ray and M the mass of the emitting atom. 

§ 2. Experimental Method 

The resonance absorption was measured by moving the source so as to destroy 
the resonance by the Doppler effect. The source was driven by a flat-topped 
saw-tooth waveform, so that it was stationary and moving for equal periods of 
time; a feedback amplifier was used to ensure that the motion of the source 
followed the driving waveform. Pulses from a scintillation counter were passed 
through a single-channel pulse height analyser and then gated into separate 
counting channels for the stationary and moving periods ; the counting rate was 
obtained by simultaneously gating pulses from a standard oscillator. The 
velocity in the moving part of the waveform was such as to shift the emission line 
off resonance by about ten half-widths. This velocity was calculated from the 
frequency of the driving waveform and the amplitude of motion (about 0-2 mm, 
measured with a micrometer-eyepiece microscope). 

To obtain maximum recoilless emission the source was maintained within a 
few degrees of liquid air temperature throughout the experiments, by enclosing 
it in a chamber surrounded by liquid air. The absorber was situated in the 
vacuum space below this liquid air vessel. It was held between graphite disks 
clamped in an aluminium ring ; the graphite disks were machined so as to preserve 
the shape of the absorber when it was melted. Palladium foils placed on either 
side of the absorber served both to absorb unwanted x-rays and as thermal 
radiation shields. Absorber temperatures above room temperature were obtained 
by electrical heating; for temperatures below room temperature the absorber 
was connected thermally to the liquid air vessel, and electrical heating was again 
used to obtain temperatures up to room temperature. The temperatures of 
source and absorber were measured by copper-constantan thermocouples in 
contact with them ; the thermocouples were calibrated in liquid nitrogen and at 
the melting point of tin. 



290 



The Mossbauer Effect in Tin 



131 



Since the temperatures of source and absorber were normally different, a 
small correction had to be made to the results to allow for the thermal shift 
reported previously (Boyle et al. 1960) ; this never amounted to more than 7%. 
The apparent absorption actually measured was less than the true absorption 
because of the finite source velocity used ; this factor was allowed for in the sub- 
sequent reduction of the results. 

§ 3. The Debye- Waller Factor 

To convert the measured absorptions into values of the Debye- Waller factor 
it was necessary to know the fraction of recoilless y-rays emitted by the source. 
To this end measurements were made with three absorber thicknesses of 
approximately 0-001 in., 0-003 in. and 0-008 in. After correcting for back- 
ground to the 24 kev photopeak passed by the single-channel analyser and for the 
thermal shift, the values of absorption as a function of absorber thickness were 
plotted on double logarithmic graph paper for six selected temperatures. These 
six sets of three points could then be compared with a calculated curve of apparent 
fraction of recoilless y-rays absorbed as a function of na e~ 2W , where n is the 
number of atoms per cm 2 , and a is the absorption cross section at resonance. In 
this way six values of the fraction of recoilless y-rays emitted by the source were 
obtained; they agreed within the experimental error and had a mean value of 
19-4%. All the results could then be expressed as fractions of recoilless y-rays 
absorbed, and using the known value of a and the measured values of «, e~ 2W 
could be found from the theoretical absorption curve. The results are shown as 
a function of temperature in Fig. 1. A slight extrapolation of these results 
yields an expected Debye- Waller factor for the source of 40% ; after correction 
for self-absorption due to finite source thickness we expect a recoilless emission 
of 23%. The presence of 113 In K x-rays in the source could account for the 
difference between this value and the directly measured value of 19-4%; this 
latter estimate is probably about 10% too low, however, because of broadening 




200 300 

Temperature (\) 



Fig. 1. Temperature dependence of the Debye-Waller factor. Broken line, Eqn (5) 
for 0=142°k; chain curve, corrected for change in due to thermal expansion; 
full curve, corrected for change in due to thermal expansion, and for the effect 
of anharmonicity at constant volume. 



291 

132 A. J. F. Boyle, D. St. P. Bunbury, C. Edwards and H. E. Hall 

of the emission line by self-absorption. Fortunately, the corresponding error 
in our values of the Debye- Waller factor for the absorber is of second order in the 
source and absorber broadenings, and is probably less than 1%. 

For T>\® the high temperature approximation to Eqn (4) is not in 
error by more than 7%, and we may put 

- 37? 2 T 

2W = kV =w (5) 

The broken line in Fig. 1 represents Eqn (5) with = 142°k. Measure- 
ments of the specific heat of tin indicate a Debye of 195 °k below 2°k, but the 
effective value of falls rapidly to a minimum of 125 °K at about 12°k. In view 
of these large deviations from the Debye law, an effective of 142°k for the 
Debye- Waller factor does not seem unreasonable. But the experimental results 
deviate more and more from the linear law of Eqn (5) as the temperature 
rises. The result x 2 ozT at high temperatures depends only on the assumption 
of harmonic forces ; an explanation of the non-linear relation between W and T 
shown in Fig. 1 must therefore be sought in terms of anharmonicity of the 
lattice vibrations. 

Anharmonicity can affect x 2 in two ways. First, the effective Debye will 
be altered by thermal expansion; this is the effect considered by Zener and 
Bilinsky (1936). If we assume that all lattice frequencies are changed in the 
same proportion by expansion 

31n0 _ VP ,,v 

aTnT~c^' {} 

where /3 is the coefficient of cubic expansion and K is the isothermal com- 
pressibility. From Eqn (6) we have 

^ln0 VjP_ 
dT C V K' {) 

and if we further assume Gruneisen's law that jSocC v 

-(I) - *©7> 

= S( r -5 e > < 8 > 

since the specific heat is almost classical in the temperature range with which we 
are concerned. If we insert numbers into Eqn (8) we find that the value of 
x 2 given by Eqn (5) has to be increased by a fraction (2-63 x 10 _4 )(r — 10); 
values thus corrected are shown by the chain line in Fig. 1. It can be seen that 
this correction is too small by a factor of more than 3 to account for the experi- 
mental results. The correction is not very certain because we have, for example, 
ignored the very large anisotropy in the thermal expansion of tin ; but it seems 
most unlikely that our estimate could be in error by so large a factor, and there is 
indeed a second mechanism by which anharmonicity can increase x 2 . Even at 
constant volume x 2 is not proportional to T at high temperatures if the forces are 
anharmonic. To estimate the magnitude of this effect consider a particle bound 
in the one- dimensional potential 

V(x) = %ax 2 + bx 3 + cx*; (9) 



292 

The Mossbauer Effect in Tin 133 

it is then easily shown that in classical conditions (cf. Peierls 1956) the specific 
heat is, including only the lowest order corrections, 

"-"■{'♦?[-©'-<©]} <■•> 

and the mean square displacement is 

-tKtW" 12 ©]}- j <"> 

This model therefore leads us to expect a fractional deviation in x 2 about three 
times that in the specific heat ; more or less according to the sign and magnitude 
of the quartic term in the potential. The measured expansion coefficient and 
specific heat of tin show that the fractional excess in C v at high temperatures, 
excluding the electronic specific heat, is moderately well represented by 

Af 

=1 = 1-38 x 10-*(T- 50). (12) 

C v 

If a correction of 4-5 times this amount is added to the previous correction to the 
value of ^calculated from Eqn (5) we obtain the full curve of Fig. 1, which 
is in excellent agreement with the experimental results. This factor of 4-5 
implies that in Eqn (9) 

5-K9'- < 13 > 

In view of the grossly over-simplified model used to derive this result too much 
significance should not be attached to it; but it is perhaps worth pointing out 
that for a 12-6 interatomic potential ac/b 2 = 1-26. 

§ 4. Diffusion near the Melting Point 

Experimental values of absorption in the 0-008 in. absorber near the melting 
point are shown in Fig. 2. The full line corresponds to the full curve of Fig. 1. 
It is clear that there is a significant decrease in resonance absorption below 
the value expected from the Debye- Waller factor in the last few degrees below 
the melting point. Measurements of line width were also made at temperatures 
approximately 0-8 and 8°k below the melting point, by taking additional readings 
with a source velocity corresponding approximately to the half- width; it was 
found that 0-8 °k below the melting point the width of the absorption line had 
increased by a factor of l'97j;J:||. 

To see how such an effect might arise, consider the effect of diffusion on the 
factor (e~*- r( ' ) e* lc - r(0) ) in Eqn (2). In an ideal lattice this tends asympto- 
tically to e~ 2W at infinite time, but if any diffusion occurs this is no longer true. 
In fact, the wavelength of 24kev y-rays is sufficiently short that if the atom has 
jumped to another lattice site at time t, (e _lkr(<) > is effectively zero. The value 
of ^e _ik r(0 e ikr(0) ) is thus the value in the absence of diffusion multiplied by the 
chance that the atom concerned has not diffused from its original lattice site, i.e. 

(e-*K0e*K0))^e- 2pr exp[-r D |f|], (14) 

where r D is the mean jump frequency of the atoms in the diffusion process. 
Eqn (2) thus becomes 



p{k) 



-If" exp [ic(k -k a )t] exp [(|r y + r D )|*|]e-^A (15) 



293 



134 A. J. F. Boyle, D. St. P. Bunbury, C. Edwards and H. E. Hall 

The effect of diffusion is thus to increase the width of the absorption line by a 
factor (r y + 2r D )/r y and thus to reduce its maximum height by the same factor. 
When this absorption line is folded with the emission line of width \T y , we find 
that the actual reduction in resonant absorption is by a factor (r y + r D )/r y . 



-w-v^- 



T-T m (°K) 



W^-^H-o- 



Fig. 2. Resonance absorption in the 0-008 in. absorber near the melting point. 
The full line corresponds to the full curve in Fig. 1 . 




Fig. 3. Diffusion near the melting point. The point marked by a circle was deduced 
from the width of the resonance absorption, and the other points from its amplitude 
(Fig. 2). 



Values of 2r D /r y calculated from the measured reduction in absorption are shown 
in Fig. 3 ; a point calculated from the change in line width is also shown ; the 
agreement with the other points provides some confirmation of Eqn (15). 
The diffusion coefficient D may be estimated from the relation for a random walk 
in three dimensions 

r D = 6D/S2, (16) 

where 8 is the step length. A tin atom has four nearest neighbours at 3-02 A, and 
two others at 3T6 A. We therefore have taken a mean value of 3T A to obtain the 
approximate values of diffusion coefficient indicated on the right-hand ordinate of 
Fig. 3. Measurements of self- diffusion in tin (Fensham 1950) only extend up to 
223- 1°k (8-8°k below the melting point), where Z>=2-65 x lO" 10 cm 2 sec" 1 along the 
tetrad axis and Z) = 0-93 x 10~ 10 cm 2 sec -1 perpendicular to it. Our results could 
reasonably be extrapolated to join these smoothly, but they indicate a greatly 
accelerated rise in diffusion near the melting point. Such an effect has been found 
by conventional methods in indium close to the melting point by Eckert and 
Drickamer (1951). This encourages us to believe that our proposed explanation of 



294 

The Mossbauer Effect in Tin 135 

the behaviour of the Mossbauer effect near the melting point is indeed correct. We 
may also mention that very rapid recrystallization of an unmelted absorber foil 
was observed at temperatures where the Mossbauer effect was reduced. 

§ 5. Conclusion 

Our experiments show that the intensity of the Mossbauer effect is essentially 
determined by the mean square displacement of the emitting atom during the 
lifetime of the excited state, in accordance with Eqn (2). The effect of 
diffusion may be thought of crudely as defining rather precisely the time at which 
the y-ray was emitted, and thereby broadening its energy. It is essentially 
because of diffusion that the effect is not observed in the liquid ; but there seems 
to be no reason in principle why it should not be observed in a liquid if the diffusion 
coefficient were sufficiently low and the lifetime sufficiently short. 

Our analysis in § 3 shows that measurements of the Debye- Waller factor 
may give useful information about lattice anharmonicity, if a more thorough 
theoretical analysis can be given. For this purpose the Mossbauer effect has the 
advantage over x-ray diffraction that it readily yields absolute values of the Debye- 
Waller factor. 

Acknowledgments 

We should like to thank Dr. S. F. Edwards, Dr. A. Herzenberg and Dr. J. O. 
Newton for a number of helpful discussion, and Mr. J. R. Rook for computing 
the theoretical absorption integrals. One of us (C.E.) is indebted to the Depart- 
ment of Scientific and Industrial Research for financial support. 

References 

Boyle, A. J. F., Bunbury, D. St. P., Edwards, C, and Hall, H. E., 1960, Proc. Phys. 

Soc, 76, 165. 
Eckert, R. E., and Drickamer, H. G., 1951,^. Chem. Phys., 20, 13. 
Fensham, P. J., 1950, Aust. J. Set. Res. A, 3, 91 ; 4, 229. 
Mossbauer, R. J., 1958, Z. Phys., 151, 124. 

Peierls, R. E., 1956, Quantum Theory of Solids, § 2.3 (Oxford: University Press). 
Van Hove, L., 1954, Phys. Rev., 95, 249. 
Zener, C, and Bilinsky, S., 1936, Phys. Rev., 50, 101. 



295 



Mossbauer Effect : Applications to Magnetism 

G. K. Wertheim 
Bell Telephone Laboratories, Inc., Murray Hill, New Jersey 



The Mossbauer effect, the resonant absorption of nuclear 
gamma rays in solids, may be used to obtain the hyperfine 
structure of Fe" in magnetic materials. Experiments are per- 
formed by observing the absorption by stable Fe 67 of the 14.4-kev 
gamma ray coming from a source which contains radioactive 
Fe" produced by the decay of Co 57 . The experiments are not 
limited to naturally iron-bearing materials; other substances can 
be studied, provided only, that small amounts of cobalt can be 
introduced into lattice sites of interest. The magnetic moments 
of the ground and first excited states of Fe S7 are known and 
make possible direct determination of the field at the iron nucleus 
once the hyperfine structure has been measured. The magnetic 
field at iron nuclei has been determined in the ferromagnetic 



transition metals (Fe 3.42X10 6 oe, Co 3.12X10 6 oe, Ni 2.80X10 5 
oe at 0°K), but no hyperfine structure has been observed down to 
4°K in the antiferromagnetic transition metals, Mn and Cr. In 
the case of yttrium-iron garnet the fields at the iron atoms in the 
two types of sites have been obtained (tetrahedral 3.9X10 5 oe, 
octahedral 4.7 X10 6 oe). The most complete analysis so far has 
been made in FeF 2 where the magnetic field in the antiferro- 
magnetic state (,Hr_o = 3.40Xl0 6 oe) and the quadrupole splitting 
in the paramagnetic state (31.2 Mc/sec) have been obtained. 
Other materials under investigation are the iron oxides and 
some ferrites, where, for trivalent iron, fields in the vicinity of 
5.0X10 6 oe have generally been found. 



INTRODUCTION 

THE realization by R. L. Mossbauer 1 that the 
nuclear recoil associated with gamma emission 
may be absent when the decaying atom is bound in a 
crystal lattice has led to a number of interesting ex- 
periments in nuclear physics. It also has considerable 
promise as a tool in solid-state physics, and in particular 
in magnetism. The connection between these usually 
unrelated fields arises from the narrowness of the line- 
width of the emitted gamma rays, which makes it pos- 
sible to resolve the hyperfine splitting of the nuclear 
energy levels. 2-4 These, of course, reflect the magnetic 
field at the nucleus as well as the electric field gradient 
tensor, both of which are of immediate interest to the 
solid-state physicist. 

The statement that the nuclear recoil is absent does 
not imply a violation of the law of conservation of 
momentum. In the recoil-free emission process the 
crystal containing the decaying atom recoils as a unit. 

■R. L. Mossbauer, Z. Physik 151, 124 (1958); Naturwissen- 
schaften 45, 538 (1958); Z. Naturforsch. 14a, 211 (1959). 

2 L. L. Lee, L. Meyer-Schutzmeister, J. P. Schiffer, and D. 
Vincent, Phys. Rev. Letters 3, 223 (1959). 

5 R. V. Pound and G. A. Rebka, Jr., Phys. Rev. Letters 3, 
554 (1959). 

* I. Ya Barit, M. I. Podgoretskii, and F. L. Shapiro, Zhur. 
Eksptl. i Teoret. Fiz. 38, 301 (1960). 



As a consequence of the large mass of the recoiling 
unit, the energy associated with the recoil is vanishingly 
small, and the nuclear gamma ray has an energy which 
is very closely equal to that of the nuclear transition. 
It is this fact (which we will discuss further below) 
that in turn makes possible the resonant reabsorption 
of the gamma ray by another atom of the same species, 
which is an essential part of all Mossbauer experiments. 
The fraction of decays which take place without 
recoil depends on the ratio of the free-atom recoil energy 
to the Debye energy kd D , the characteristic energy of a 
phonon. An atom in the solid may be thought of as being 
free to emit zero, one, or many phonons, but as being un- 
able to recoil with an arbitrary energy. 6 (This statement 
is true only as long as the recoil energy is sufficiently 
small so that the atom is not displaced from its lattice 
site by the recoil.) When many decay processes are 
considered it is found that the average energy of the 
emitted phonons per decay is still the free-atom recoil 
energy. From this it is immediately apparent that when 
the free-atom recoil energy is smaller than kBo, there 
will be a high probability of emitting no phonon at all. 
These facts are contained in the familiar Debye-Waller 

6 For a fuller discussion see : W. M. Visscher, Ann. Phys. (N. Y.) 
9, 194 (1960); H. J. Lipkin, ibid. 9, 332 (1960); K. S. Singwi and 
A. Sjolander (to be published). 



296 



111S 



MOSSBAUER EFFECT: APPLICATIONS TO MAGNETISM 



factor expressing the fraction of recoil-free events 
3 ErV 2/tT\ 2 1 



/=exp 



2ke D i 3\e D /]\ 



The usefulness of the zero-phonon gamma rays arises 
from their unusually narrow linewidths. The natural 
linewidth of a gamma ray is determined by the widths 
of the states involved in the decay process. Here we 
will consider only decays from an excited state to the 
ground state of a stable isotope, in which the width is 
entirely determined by the excited state. The natural 
width of the excited state is determined through the 
uncertainty principle by the lifetime of the state; a 
lifetime of 10 -7 sec, characteristic of the widely used 
Mossbauer isotope Fe 57 , leads to a level width of 
4.6X10 -9 ev (equivalent to 1.13 Mc/sec), which is 
smaller than characteristic hyperfine or quadrupole 
interaction energies in many solids. 

Other sources of line broadening must also be con- 
sidered. The thermal motion of the emitting atoms could 
be a serious limitation in a gaseous or liquid source, but 
since the zero-phonon process takes place to a measur- 
able extent only in solids, we will consider only this 
case. Lattice vibration frequencies are characteristically 
10 13 sec -1 . If an isotope is considered whose excited state 
has a lifetime long compared to the period of the lattice 
vibration, there will be no first-order Doppler broadening 
or shift from this cause. There will, however, be a second- 
order Doppler shift* 57 which depends only on the average 
of the square of the velocity, i.e., on the kinetic 
energy, of the lattice atoms. This shift is small and does 
not affect the resolution since it is of the same magnitude 
and in the same direction for all emitting atoms. Broad- 
ening can also arise if the environment of the emitting 
atoms varies, as it might in an alloy, or if the crystalline 
fields at the emitting atoms have frequencies compar- 
able to the lifetime of the excited nuclear state. 

The existence of recoil-free emission and resonant 
absorption is best demonstrated in a simple transmission 
experiment in which gamma rays from a source pass 
through an absorber to a detector. If the resonant 
absorption is destroyed, the counting rate at the detector 
will increase. This is most simply accomplished by giving 
the source a velocity sufficient to Doppler-shift the 
energy of the emitted gamma rays by more than their 
natural linewidth; velocities of the order of a small 
fraction of 1 cm/sec are required. A simple extension 
of this idea produces a "Mossbauer spectrometer" with 
which an absorption spectrum is obtained simply by 
observing the counting rate at the detector as a function 
of the Doppler velocity of the source. 

After the original demonstration of resonant absorp- 
tion by Mossbauer using the isotope Ir 191 , and its sub- 

8 R. V. Pound and G. A. Rebka, Jr., Phys. Rev. Letters 4, 227 
(1960). 
7 B. D. Josephson, Phys. Rev. Letters 4, 341 (1960). 



sequent verification by a number of other groups, 28 
attention has shifted to other suitable isotopes. One 
of the first to be used, and one which remains of par- 
ticular interest for solid-state work, is Fe 57 . 9-12 This 
isotope offers a combination of desirable properties 
which have already made possible experiments ranging 
from a verification of the gravitational red shift to the 
determination of magnetic fields in solids. The cobalt 
parent of Fe 87 has a 270-day half -life, convenient for 
most work. The decay is by electron capture, and the 
neutrino which accompanies this process has an energy 
of ~0.6 Mev. The iron is left in a fairly low state of 
excitation (134 kev), from which it makes a 120-kev 
gamma transition to the first excited state whose life- 
time is 10~ 7 sec. The low energy (14.4 kev) of the tran- 
sition to the ground state used in the experiment leads 
to a large fraction of zero-phonon decays. In addition, 
Fe 57 has the advantage that iron is a component of 
many interesting magnetic materials. 

EXPERIMENTAL 

Experiments always involve a source which contains 
the radioactive species, an absorber which contains the 
stable isotope, and a radiation detector (Fig. 1). The 
substance under study may be used as either the source 
or the absorber. If a substance is naturally iron-bearing 
there are a number of advantages to using it as an ab- 
sorber. One of these is that the iron is in a normal 
lattice site, rather than in a site characteristic of cobalt ; 
another is that there is no preceding electron-capture 
decay, which could result in a displaced or multiply 
ionized atom. The concentration or iron in the material 
should be high enough so that an absorber of areal 
density 0.1 mg/cm 2 of Fe 57 can be made. Since the 
natural abundance of Fe 57 is only 2.14%, the use of 
enriched isotope may be desirable. (It has also been 
possible to use as absorbers substances not normally 
iron-bearing into which Fe 57 obtained as separated 
isotope was introduced in amounts less than 1 atom- 
percent.) 

Substances which do not normally contain iron may 
be studied by incorporating small amounts of Co 57 into 
them and using them as sources. Such studies yield 
information on the fields at isolated impurity atoms. 
This general approach is of course also applicable to 
iron-bearing substances. In the case of metallic iron no 
difference has been found between experiments using the 
iron as an absorber in the pure form and those using 
it as a source containing trace amounts of Co 67 . In 
insulators, however, major differences between these 
two types of experiment have been observed. 

8 P. P. Craig, J. G. Dash, A. D. McGuire, D. Nagle, and R. D. 
Reiswig, Phys. Rev. Letters 3, 221 (1959). 

9 R. V. Pound and G. A. Rebka, Jr., Phys. Rev. Letters 3, 439 
(1960). 

10 J. P. Schiffer and W. Marshall, Phys. Rev. Letters 3, 556 
(1959). 

11 S. S. Hanna et al., Phys. Rev. Letters 4, 28 (1960). 

" G. DePasquali, H. Frauenfelder, S. Margulies, and R. N. 
Peacock, Phys. Rev. Letters 4, 71 (1960). 



297 



G. K. WERTHEIM 



112S 




-SOURCE 



3""0— -o-ooi' AT. 




Fig. 1. Experimental arrangements used by the author: (a) 
Stationary source and moving, unsplit absorber, (b) Moving, 
unsplit source and stationary absorber. 

One of the components of the experiment, either the 
source or the absorber, should have an unsplit line in 
order for the hyperfine spectrum of the other to be ob- 
served directly. A number 'of substances have already- 
been used for this purpose. The first to be proposed were 
the ferrocyanides, 13 which are diamagnetic and have 
no hyperfine field at the iron atom. In general, these 
suffer from the disadvantage that they do not contain 
much iron for use as absorbers. Considerable improve- 
ment can be obtained with a ferrocyanide made with 
separated Fe 67 isotope. 14 Even at room temperature this 
substance, used as an absorber, is by far the most suc- 
cessful we have yet used, the criteria of excellence being 
the linewidth and the ratio of resonant absorption to 
photoelectric absorption. Unfortunately, all attempts to 
make sources by incorporating Co 57 into a ferrocyanide 
have failed 13 because the structure of the cobaltocyanide 
is different from that of the ferrocyanide. 

13 S. L. Ruby, L. M. Epstein, and K. H. Sun, Rev. Sci. Instr. 
31, 580 (1960). 

14 The author is indebted to J. D. Struthers of this laboratory 
or the preparation of this material. 



A second group of substances that has been widely 
used is the stainless steels. 15 Most of the experiments 
described in this paper were done with sources made by 
incorporating Co 57 into type 310 stainless. Here it has 
proved to be easy to prepare sources by simply evapo- 
rating Co 57 Cl2 solution to dryness on stainless steel and 
diffusing at 950°C in a carefully evacuated quartz 
capsule. 

The linewidth of the absorption or emission line in 
stainless steel has been examined in some detail, and 
has generally been found to be considerably larger than 
the natural linewidth. Any or all of the following mech- 
anisms may contribute. 

(1) The environment of the various iron atoms in 
an alloy is necessarily different. Since it is known that 
the chemical environment can shift the energy of the 
nuclear levels, it is possible that there is inhomogeneous 
broadening due to this effect. 

(2) The existence of a single line of natural width is 
predicated on an electron spin correlation time suf- 
ficiently short to destroy the hyperfine interaction. For 
a broadening of less than 10%, a spin correlation time 
less than 3X 10~ 12 sec is required. The actual correlation 
time is not known, but values of 10~ 12 or 10~ 13 sec do 
not seem unreasonable. Some indication of a tempera- 
ture-dependent linewidth, which would confirm that 
this mechanism is operative, has been obtained. 

(3) Broadening can also arise from the finite thick- 
ness of the source or absorber, and begins to be ap- 
preciable (30%) when the radiation traverses one ab- 
sorption length of material. The importance of this 
effect in stainless steel of the thickness generally used 
as an absorber has been investigated. A series of ex- 
periments have been conducted using as a source Co 57 
diffused into chromium metal, the combination which 
produces the narrowest unsplit line found so far, and 
using as absorbers type 310 stainless in a variety of 
thicknesses. The linewidth was 0.045 cm/sec, or more 
than two times the natural linewidth. No difference 
in linewidth was found with absorbers ranging from 
0.00025 in. to 0.0010 in., the latter being the thickness 
usually used in experiments. It was noted that the frac- 




-0.4 -0.2 o 0.2 0.4 
VELOCITY IN CM/SEC 

Fig. 2. The hyperfine spectrum of Fe" in iron metal, obtained 
with a stainless steel source and a natural iron absorber 0.001 in. 
thick. 



' G. K. Wertheim, Phys. Rev. Letters 4, 403 (I960). 



298 



.13S 



MOSSBAUER EFFECT: APPLICATIONS TO MAGNETISM 



tional absorption in these experiments was not propor- 
tional to absorber thickness. This fact might be con- 
sidered evidence that the absorber is thick, i.e., longer 
than one absorption length, but the effect could equally- 
well be caused by the increase in the degraded 120-kev 
radiation falling into the energy selection channel. The 
deterioration of the gamma-ray scintillation spectrum 
with increasing absorber thickness is in accord with the 
latter interpretation. 

These difficulties, inherent in the use of an alloy, have 
not prevented the widespread use of stainless steel, but 
there is a continuing interest in other substances. One 
of the most promising is KFeF 3 , a cubic material with 
perovskite structure, used as an absorber at room tem- 
perature where it is paramagnetic. Its spin correlation 
time at room temperature is short enough to result 
in a relatively narrow line, while its cubic nature assures 
the absence of quadrupole interaction. Its use as a 
source is being studied. 

Experimentally, two distinct methods for taking data 
have emerged. The first uses motion at a constant 
velocity, usually symmetrical in the forward and back- 
ward directions; different velocities are obtained by 
changing frequency or amplitude. As the source of 
motion, electromechanical transducers, cams, lathes, 
and constant-velocity servos have been used. The ex- 
periments reported here were done with a system of 
this type employing a loudspeaker voice coil driven 
at constant velocity by dc coupled transistors from the 
symmetrical sawtooth wave output of a Hewlett- 
Packard function generator. In the second method, all 
desired velocities are included in the motion, and the 
counts are sorted according to instantaneous velocity 
of the source relative to the absorber. This is generally 
done by using a sinusoidal motion, or more advantag- 
eously a double parabola, which has the feature that 
equal time is spent at each velocity increment. The 
instantaneous velocity is determined by a pickup coil 
rigidly attached to the moving source and placed in a 
uniform magnetic field. The resulting signal is either 
fed directly into the address logic in a multichannel 
analyzer or else used to modulate the amplitude of the 
energy-selected counts, which can then be sorted by an 
unmodified multichannel analyzer. 

The first method places somewhat higher demands on 
counting rate stability and requires a motion which is 
more difficult to obtain, especially at high velocities. 
On the other hand, it is more flexible, since part of an 
absorption spectrum can be examined without sweeping 
through the whole range. The second method requires 
a normalization of the data if a sinusoidal motion is 
used, since more time is spent at high velocities than 
near zero. However, this problem is absent when a 
double parabola is used, and for many experiments this 
may be the most attractive approach. It might be 
noted, however, that the multichannel analyzer system 
does not collect data any faster than the constant 
velocity system. 



EFFECTIVE MAGNETIC FIELDS IN THE 
TRANSITION METALS 

The first substance in which the field at an iron atom 
was determined by the Mossbauer effect was metallic 
iron itself. 16 The experiment was originally done with 
an iron source as well as an iron absorber. In addition 
to the field at the nucleus (3.33X 10 6 oe at 300°K), 
the magnetic moment of the first excited state was ob- 
tained. In subsequent experiments in which an external 
magnetic field was superposed, it was shown that the 
direction of the field at the nucleus is opposite to the 
external magnetization, 17 a result contrary to theoretical 
expectations. 18 

The use of an iron source and absorber gives rise to 
an extremely complicated pattern. If a source emitting 
an unsplit line is used instead, the hyperfine pattern 
of Fe 57 can be obtained directly (Fig. 2). This type of 
experiment can readily be adapted to determine the 
field at an iron impurity atom in the other transition 
metals. 15 Such experiments have actually been done by 
using an unsplit absorber, such as stainless steel, and 
making a source by diffusing the radioactive species, 
Co 67 , into rolled foils of the material under study. 19 Re- 
sults for cobalt and nickel at room temperature are 
shown in Fig. 3. The fields at room temperature are 
found to be3.10±0.05X10 5 oe and 2.65±0.05X10 6 oe; 
when these are extrapolated to 0°K using the known 
magnetization curve, values of 3.12±0.05X10 B oe and 
2.80±0.05X10 5 oe for cobalt and nickel, respectively, 
are obtained. 



Si 

i 

z 

8 

o 

2 250 



370 
360 

3.S0 


^ 


r 


y 


»v 


V 


*\ 


I 

1 


Fe 57 in Co 

Y 




-O.B -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 

VELOCITY IN CM/SCC 

Fig. 3. The hyperfine structures of impurity Fe 57 in cobalt and 
nickel metal, obtained with a stainless steel absorber 0.001 in. 
thick. Radioactive Co" was diffused into thin cobalt and nickel 
foils which were then used as sources. 



16 S. S. Hanna et al., Phys. Rev. Letters 4, 177 (1960). 

17 S. S. Hanna et al., Phys. Rev. Letters 4, 513 (1950). 

18 W. Marshall, Phys. Rev. 110, 1280 (1958). 

18 The foils of type 310 stainless steel, high-purity iron, and 
nickel were supplied by K. M. Olsen of this laboratory. 



299 



G. K. WERTHEIM 



114S 



Table I. Magnetic fields (in units of 10 6 oe) at transition 
metal nuclei located in transition metals. 



Nuclei at which 
field is measured 



Lattice atoms 
Co 



Ni 



3.42" 
3.20" 



3.12 b 
2.20° 
2.134 d 



2.80 b 
0.80° 



1 See footnote 20. 



These values should be compared with the results 
obtained for cobalt and nickel atoms in the transition 
metals, using a variety of other techniques. The fields at 
cobalt in a range of Fe-Co and Ni-Co alloys were deter- 
mined by Arp, Edmonds, and Petersen 20 from the 
measurement of the nuclear contribution to the heat 
capacity. Their results gave a value of 3.20X 10 5 oe at a 
cobalt atom in iron and 2.20X 10 6 oe at a cobalt atom in 
cobalt. An independent measure of the field at a cobalt 
atom in cobalt has also been obtained by Gossard and 
Portis 21 using nuclear resonance techniques. For fee 
cobalt, they found a value of 2.134X 10 5 oe at 0°K. Most 
recently a value for the field at a nickel atom in nickel 
has been similarly determined. 22 These results are sum- 
marized in Table I. 

It may be noted that the field at the iron nucleus in 
a given metal is in every case larger than the field at 
the host lattice nucleus, and that fields at all the iron 
nuclei are of similar magnitude. This suggests that the 
field at an iron nucleus is due largely to its own electrons 
and depends only slightly on the host lattice magnetiza- 
tion, an idea which is borne out by a recent experiment 
on a CoPd alloy, 23 in which isolated impurity iron atoms 
were found to have fields similar to those in other 
metals studied. This property of the iron atom indicates 
that it is not a good field probe, and to some extent les- 
sens the interest in impurity experiments. However, 
the present results do help to elucidate the various 
contributions to the field at the nucleus in a ferromagnet, 
and for that reason remain of interest. 

In spite of this limitation on the utility of iron atoms, 
some interesting results have been obtained by the ex- 
tension of experiments of the type described above to 
the antiferromagnetic transition metals, manganese and 
chromium. These experiments were quite analogous to 
those for the ferromagnetic case. The source was made 
by diffusing Co 57 into electrolytically deposited Cr and 
Mn ; the absorber was isotopically enriched potassium 
ferrocyanide. 



20 V. Arp. D. Edmonds, and R. Petersen, Phys. Rev. Letters 
3, 212 (1959). 

21 A. C. Gossard and A. M. Portis, Phys. Rev. Letters 3, 164 
(1959) ; A. M. Portis and A. C. Gossard, J. Appl. Phys. 31, 205S 
(I960). See also W. A. Hardy, J. Appl. Phys. 32, 122 (1960), 
this issue. 

22 L. J. Bruner, J. I. Budnick, and R. J. Blume, (to be published). 

23 D. E. Nagle et al., Phys. Rev. Letters 5, 364 (1960). 



It would not be surprising to find that the iron ex- 
hibits a splitting characteristic of a field of 3X10 8 oe 
when the host lattice is below its Neel temperature, but 
actually, in the case of chromium, data taken between 
4°K and 370°K show an unsplit line whose width de- 
creases uniformly with increasing temperature (Fig. 4) 
without discontinuity of any sort even at 308°K, the 
Neel temperature. The data for a-manganese are similar, 
except that the change in linewidth is much less pro- 
nounced. These results suggest that the exchange 
coupling between iron and chromium or manganese is 
sufficiently small so that the iron atoms are not aligned 
by the host lattice atoms. This conclusion is consistent 
with the results obtained from low-temperature sus- 
ceptibility measurements in dilute iron-chromium al- 
loys, 24 which also indicate that the iron is not aligned 
by the antiferromagnetic Cr system. 

FERROUS FLUORIDE 

As an illustration of the application of the Mossbauer 
effect to a naturally iron-bearing antiferromagnet, let 
us consider the determination of the hyperfine structure 
of Fe 57 in ferrous fluoride. 26 This material is antiferro- 
magnetic, with a transition temperature of 79°K. It 
has the rutile structure ; thus the symmetry around the 
iron atoms is no higher than that characterized by 
three mutually perpendicular reflection planes. The 
electric field gradient (EFG) tensor at the iron atoms 




-0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 
VELOCITY IN CM/SEC 

Fig. 4. The line shape of the 14.4-kev gamma ray of Fe 67 , 
produced by the decay of Co 67 diffused into chromium metal. 
The data were taken with an absorber of potassium ferrocyanide 
made with enriched Fe 57 . 



24 M. M. Newman and K. W. H. Stevens, Proc. Phys. Soc. 
(London) 74, 290 (1959). 
26 G. K. Wertheim, Phys. Rev. 121, 63 (1961). 



300 






115S 



MOSSBAUER EFFECT: APPLICATIONS TO MAGNETISM 



therefore does not have axial symmetry and may be 
characterized by two independent parameters, usually 
chosen to be the largest diagonal element of the tensor 
and a parameter which describes the deviation from 
axial symmetry. 

The experiment was done using two 0.005-in. thick 
single-crystal slabs of the material as absorbers. 26 One 
slab had the c axis normal to its plane, the other parallel. 
A radioactive source, consisting of Co 57 diffused into 
type 310 stainless steel, was mounted on a loudspeaker 
voice coil driven with a constant-velocity sawtooth wave 
as described above. 

In the paramagnetic state, at room temperature, two 
well-defined absorption lines were obtained [Fig. 5(a)]. 
Their spacing indicates a quadrupole splitting of 31.2 
Mc/sec due to the interaction of the excited state quad- 
rupole moment with the crystalline electric field grad- 
ient. A large displacement of the centroid of the ab- 
sorption from zero velocity, i.e., the energy of the 
gamma ray emitted by iron atoms in stainless steel, is 
also observed. Only a small part of this displacement 
can arise from a difference in the second-order Doppler 
shift 6 in the two materials. The major part is due to a 
nuclear isotope shift 27 arising both from the removal 
of 4s electrons in going from metallic binding in stainless 
steel to ionic binding in ferrous fluoride, and from the 
exchange polarization of the inner 5 electrons by the 
d shell, which causes a charge rearrangement of the 
.y electron wave function at the nucleus. 28 The effect 
arises from the electrostatic interaction of the nucleus 
with the electronic wave functions, 27 and is observable 
provided that the strength of the interaction is different 
for the ground and the excited states of the nucleus, 
so that the corresponding energy levels are shifted by 
different amounts. This will be the case if the nuclear 
size is different in the two states. 

In the antiferromagnetic state, six hyperfine compo- 
nents are resolved [Fig. 5(b),(c)]. (In the case where 
the magnetic c axis is prependicular to the plane of 
the absorber two lines, corresponding to the Am = Q 
transitions, are missing. This is in accord with the radi- 
ation pattern for these transitions, which has zero in- 
tensity in the direction of the axis of quantization.) The 
spacing of these lines may be analyzed in terms of ex- 
cited and ground-state splitting. The energy levels of 
the excited state are given as functions of three param- 
eters, the magnetic hyperfine interaction gf3H, the z 
component of the electric field gradient tensor eq, and 
an asymmetry parameter 



/dw dW\ /dw 

\dv 2 dx 2 // dz 2 



26 The oriented slabs of ferrous fluoride were obtained from 
V. Jaccarino. 

27 O. C. Kistner and A. W. Sunyar, Phys. Rev. Letters 4, 412 
(1960). 

28 V. Heine, Phys. Rev. 107, 1002 (1957); W. E. Blumberg and 
V. Jaccarino (private communication). 



| 15,000 

m 18,000 



a 1 6,000 

</) 

l- 

| 15,000 

O 



- 


o 


\ 


f%t 


^T 


Otffe 






Y 


f y 


290° K 




1 1 1 


V 


V 

1 


1 1 1 







PERPENDICULAR 


- *lftfc 




^2o >- u o u <v y^ 


1 


1 1 1 


wlf 45 ° K ^& 

1 1 1 1 





^^s^X 


f^^xK 




iofl. *? 


fSc 


17,000 


> 


\T^ 


f?y 


rv^v 




16,000 






v 


PARALLEL 
45°K 




15,000 


i i 


i i 


1 


1 1 1 


1 



-0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 

VELOCITY IN CM/SEC 

Fig. 5. Resonant absorption in 0.005-in. single-crystal ferrous 
fluoride absorbers (a) above the Neel temperature; (b) below 
the Neel temperature with magnetization along the direction of 
observation; and (c) below the Neel temperature with magneti- 
zation perpendicular to the direction of observation. 



as follows : 

e 2 Qq[ / 



1+- 



4gffff \ 2 , ifV 
e 2 Qq 



HI 



- / — [(l- 



4g/3ff \ 2 iTl* 
e 2 Qq ) 3 J ' 



The analysis yields two independent measures of the 
magnetic field at the iron nucleus, in terms of the ground 
and excited state moments, respectively. The average 
of these measurements gives a value of 3.13X 10 5 oe at 
45°K, the temperature of the measurements. Extra- 
polated to 0°K this becomes 3.40X10 5 oe. The quad- 
rupole interaction deduced from the antiferromagnetic 
state data is consistent with that determined from the 
paramagnetic state, but it is found in addition that the 
direction of the major axis of the EFG tensor is perpen- 
dicular to the axis of magnetic alignment and that the 
asymmetry parameter is 0.33. 

YTTRIUM-IRON GARNET 

One of the more complex systems which has recently 
been investigated 29 is yttrium-iron garnet, Y 3 Fe 2 (Fe0 4 )3. 
The structure of this material is well known, 30 and for 
the present purposes it suffices to point out that iron 
atoms are located in two nonequivalent sites, those 
with tetrahedral symmetry and those with octahedral 

29 This work was done in collaboration with Miss C. Alff of 
Columbia University, during a summer appointment at this 
laboratory. A fuller account will be published. 

30 S. Geller and M. A. Gilleo, J. Phys. Chem. Solids 3, 30 (1957) ; 
S. Geller and M. A. Gilleo, Acta Cryst. 10, 239 (1957) ; F. Bertaut 
and F. Forrat, Compt. rend. 242, 382 (1956). 



301 



K. WERTHEI M 



116S 



o 127 

w 126 

O 125 
o 
co 124 

z 123 

{2 122 

§ 121 - 



xlO* 




MAGNETIC FIELD IN 
(III) DIRECTION 



VI 




TETRAHEDRAL (0=55°) 



OCTAHEDRAL (e * 70°) 



Table II. Characteristics of iron sites in yttrium-iron garnet 
for the two cases used in the experiment. 



■1.0 -0.6 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1.0 
VELOCITY IN CM /SEC 

:rystal 

yttrium-iron garnet absorber with magnetization in the [111] 
direction. The source was stainless steel. 



symmetry. The relative number of these is 3:2. Both 
sites have axes of symmetry sufficiently high to assure 
that the electric field gradient tensors are axially sym- 
metric. The tetrahedral sites have fourfold rotary in- 
version axes in [100] directions, while the octahedral 
sites have threefold axes in [111] directions. The [111] 
directions are the directions of easy magnetization, but 
in a thin slab without an externally applied field the 
magnetization tends to be normal to the surface. 

The experiments were performed using as an ab- 
sorber a thin slab of the material cut normal to a [110] 
direction from a single crystal grown by Nielsen of this 
laboratory. 31 The slab was cut and lapped to a thickness 
of 0.002 in. according to the process described by 
Dillon. 32 Data were taken at room temperature with a 
magnetic field applied in either the [111] or the [100] 



;I0' 



{^ 157 
o '56 

8 |55 

J 154 
«o 153 
| 152 
8 151 
150 




MAGNETIC FIELD IN 
(100) DIRECTION 




TETRAHEDRAL {6 '90') 



OCTAHEDRAL (©=55°) 



TETRAHEDRAL (0»O°) 



1.0-0.8 -0.6 



0.4-0.2 0.2 Q4 
VELOCITY IN CM/SEC 



0.6 0.8 1.0 



Fig. 7. Resonant absorption in the absorber of Fig. 6 with 
the magnetization in the [100] direction. 



31 The YIG was obtained from J. F. Dillon. 

32 J. F. Dillon and H. E. Earl, Am. J. Phys. 27, 201 (1959). 



Field direction 
111 
Angle" Intensity 6 



100 
Angle* Intensity b 



Tetrahedral 54°44' 



Octahedral 70°32' 

0° 



90° 
0° 

54°44' 



■ Angle between direction of magnetization and axis of electric field 
gradient tensor. 

b Relative number of iron atoms occupying sites characterized by the 
given angle. 



direction in the plane of the slab. It should be pointed 
out that when the field is in the [111] direction the 
symmetry axes of all the tetrahedral sites make the 
same angle (54°44') with the applied field, while the 
axes of the octahedral sites make angles of either 0° 
or 70°32', in the proportion of 1 to 3. Similarly, when 
the field is in the [100] direction, the [111] symmetry 
axes of all the octahedra make an angle of 54°44' with 
the field, while the axes of the tetrahedra make angles 
of 0° or 90° in the proportion of 1 to 2. (This informa- 
tion is summarized in Table II.) Three hyperfine pat- 
terns are thus to be expected in each experiment. 

The experimental results obtained at room tempera- 
ture are shown in Figs. 6 and 7. In the [100] case (Fig. 
7) all three hyperfine spectra are resolved, while in the 
[111] case (Fig. 6) only the two stronger ones are seen. 
The identification of lines belonging to a given hyperfine 
spectrum was made partially on the basis of the intensi- 
ties of the absorption lines and partially on the basis 
of some simple properties of the hyperfine patterns. 
The resulting groupings were in every case unambiguous. 

The magnetic fields at the iron nuclei in each case 
were obtained directly from the distance separating 
the second and fourth or third and fifth absorption lines 
in the pertinent hyperfine spectrum. This distance de- 
pends only on the ground-state magnetic moment, and 
is independent of the quadrupole coupling. Values were 
obtained of 3.92±0.05X10 5 oe for the tetrahedral sites 
and 4.74±0.06X10 5 oe for the octahedral sites. The 
agreement between the independent determinations 
made for the [100] and [111] cases was satisfactory. 

Further analysis was based on a direct comparison 
of the experimental hyperfine spectrum with spectra 
computed for the known angles between the magnetic 
field and the axis of the EFG tensor. The computations 
were based on the tabulations of Parker, 33 which are 
applicable to the excited state and give the hyperfine 
splitting as a function of the parameter \ = e 2 Qq/^y.H 
measuring the strength of the quadrupole coupling rela- 
tive to the magnetic hf coupling. The best value for X 
was 0.10±0.02 in each case. The large uncertainty arises 
from a weak dependence of the hfs on X, which is par- 
ticularly pronounced in the 54°44' case. The resulting 

83 P. M. Parker, J. Chem. Phys. 24, 1096 (1956). 



302 



117S 



MOSSBAUER EFFECT: APPLICATIONS TO MAGNETISM 



quadrupole couplings, expressed as the quadrupole 
transition energies in the absence of a magnetic field, 
are 9 and 11 Mc for the tetrahedral and octahedral 
sites respectively, with an uncertainty of 20%. 

Further experiments are being done to obtain the 
temperature dependence of the field at the iron nucleus. 

OXIDES AND FERRITES 

Of the iron oxides only the simplest, Fe 2 3 , in which 
all the atoms are in equivalent sites, has so far produced 
conclusive results. Kistner and Sunyar 27 have shown 
that the field at the iron atom is 5.15X10 6 oe at room 
temperature; a small quadrupole component has also 
been observed. 

Stoichiometric FeO probably does not exist because 
of the strong tendency of iron to be trivalent. Attempts 
to prepare this compound by the decomposition of 
ferrous oxalate in an inert atmosphere have produced 
a material which exhibits two absorption lines of un- 
equal intensity at room temperature. These may be 
identified with di- and trivalent iron, since their un- 
equal intensity and the fact that FeO is cubic rule out 
quadrupole splitting as a possible interpretation. 

In Fe 3 4 there exist both A sites, with tetrahedral 
oxygen coordination, and B sites, with octahedral oxy- 
gen coordination. Moreover, the iron occurs with two 
valences which are distributed over the sites in such a 
way that three types of iron are found in equal concen- 
tration : trivalent iron in A sites, trivalent iron in B sites, 
and divalent iron in B sites. As a result, three separate 
six-line hyperfine patterns are to be expected. In prac- 
tice these patterns appear superposed in such a way 
that analysis is at best difficult. However, it is clear 
from the data in Fig. 8 that the field at the nucleus is 
approximately 5X10 5 oe in every case. 

Among the ferrites the simplest results should be 
obtained from those with a structure like that of nickel 
ferrite ; that is, like FeO • Fe 2 3 in which all the divalent 
iron atoms are replaced by divalent atoms of another 
metal and all the remaining iron is trivalent. Results 
obtained with a thin polycrystalline absorber of 
NiO-Fe 2 3 34 indicate that the field at the iron nucleus 
is about 5.1 X10 5 oe, but the linewidths are found to 
be broad compared to those of the simple oxides. A 
number of ferrites have also been studied by Kistner and 
Sunyar 35 with generally similar results. 

CONCLUSIONS 

The Mossbauer effect of Fe 57 has proved to be a 
useful tool for the investigation of magnetic fields and 
electric field gradients at iron nuclei in ferromagnetic 
and antiferromagnetic materials. The fact that the 



!s VYvrvv w 

Fe.O, 




34 The nickel ferrite was prepared by F. J. Schnettler of this 
laboratory. 
36 O. C. Kistner and A. W. Sunyar (unpublished). 



-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.6 1.0 1.2 
VELOCITY IN CM/SEC 

Fig. 8. Resonant absorption at room temperature in powdered 
iron oxides and polycrystalline nickel ferrite. The source is 
stainless steel. 



field at an iron atom depends largely on its own elec- 
tronic shell and only weakly on its environment is 
itself interesting but lessens the usefulness of isolated 
iron atoms as field probes. A few generalizations have 
begun to emerge from the results now available: (1) the 
field at iron nuclei in ferromagnetic metals is close to 
3X10 5 oe. (2) The field in oxides, ferrites, and yttrium 
iron garnet at trivalent iron atoms in octahedral co- 
ordinations is in the vicinity of 5X 10 5 oe. These general- 
izations are as yet tentative and subject to further 
confirmation. 

While Fe 57 is without doubt the most attractive iso- 
tope for Mossbauer effect studies, and the only one 
considered in this paper, there are a number of others 
which will make it possible to extend the range of 
materials which can be investigated. The most useful 
of these is Sn 119 , an isomer with a 250-day half-life and 
a linewidth only five times greater than that of Fe 57 . 
Other possible isotopes are Dy 161 , W 182 , Ir 193 , and Au 197 . 
The maximum absorption that can be expected with 
these is very much smaller, because of higher gamma 
energy or lower Debye temperature than Fe 57 , and the 
linewidth is larger by a considerable factor. 

ACKNOWLEDGMENTS 

Conversations with V. Jaccarino, J. F. Dillon, W. E. 
Blumberg, S. Geschwind, and a number of other mem- 
bers of this laboratory have been of great assistance to 
the author by providing many valuable ideas and fruit- 
ful insights into the field of magnetism. The cooperation 
of Mrs. M. H. Read and W. M. Augustyniak in specific 
phases of the work, and the general technical aid of 
D. N. E. Buchanan are also gratefully acknowledged. 



303 



Volume 4, Number 10 



PHYSICAL REVIEW LETTERS 



May 15, I960 






DIRECTION OF THE EFFECTIVE MAGNETIC FIELD AT THE NUCLEUS IN FERROMAGNETIC IRONt 

S. S. Hanna, J. Heberle, G. J. Perlow, R. S. Preston, and D. H. Vincent 

Argonne National Laboratory, Argonne, Illinois 

(Received April 28, 1960) 



In a recent experiment 1 it was shown that in 
ferromagnetic iron the effective magnetic field 
at the iron nucleus is strongly correlated with 
the magnetization. The sense of the correlation, 
however, was not determined, i.e., it was not 
known whether the effective field was parallel or 
antiparallel to the magnetization. The sense of 
the correlation has now been established by ob- 
serving the change in the hyperfine splitting of 
the nuclear energy levels of Fe 57 on application 
of an external field of 17 to 20 koe. 

In an earlier paper 2 we presented the hyperfine 



spectrum obtained in the resonant Mossbauer 3 
absorption in Fe 57 . The interpretation given to 
the spectrum has since been confirmed in detail. 
Several groups have shown the correctness of 
the hyperfine pattern by observing the spectrum 
when different alloys and compounds of iron are 
used. 4 Gossard, Portis, and Sandle 5 have ob- 
served the nuclear magnetic resonance in the 
ground state of Fe 57 at a frequency corresponding 
to a value of the effective field in close agreement 
with the value of 333 koe deduced in reference 2. 
In addition, Ewan, Graham, and Geiger 6 have 



513 



304 



Volume 4, Number 10 



PHYSICAL REVIEW LETTERS 



May 15, I960 



found that the £2 admixture in the M radiation 
is less than 10 " 4 , which confirms that the effect 
of E2 radiation in the spectrum is indeed negli- 
gible. 2 

Experimentally it was feasible to apply a large 
magnetic field only to the source of the resonant 
radiation. The absorber was either in the fring- 
ing field of the electromagnet holding the source 
or in a small parallel magnetic field of its own, 
applied to produce a definite magnetization in the 
absorber. At the top of Fig. 1 is shown the ve- 
locity spectrum which is applicable if the hyper- 
fine splittings in source and absorber are identi- 
cal. The intensities are appropriate to the emis- 
sion of polarized radiation from the source but 
to an unpolarized absorption process. If, on the 
other hand, the hyperfine splittings in the emitter 
are about 10% greater (for example) than those 
in the absorber, then the complex spectrum at 
the bottom of Fig. 1 is obtained. It is clear that 
a study of the singlet line 6 affords the best means 
of determining the change that an external field 
produces in the hyperfine splitting. 

For the effective field at the nucleus we write 



H « = #no M o + H ext> 



(1) 



where M is a unit vector along the direction of 
magnetization in a ferromagnetic domain, and 
H no is the magnitude of the effective field in the 
absence of the external field H ex t- The latter 
quantity includes the demagnetizing field which 
is negligible for the planar samples used. Since 
Hext/^no << * in tne present experiment, it is 






Speed -* 








II 




1 

6 






s 4 : 


> 






> 







'I '"' T li f 



FIG. 1. Theoretical absorption spectra of 14.4-kev 
resonance radiation from Fe 5T . Top: metallic source 
and absorber with identical hyperfine splittings. Bot- 
tom: same source and absorber but with the splitting 
in the source increased by 10%. 



assumed in Eq^ (1) that H no is not appreciably 
influenced by H ex t. The quantity of interest is 
the sign of H no . Since M,, and H ex t are parallel 
under saturation conditions, the sign can be de- 
termined by observing whether the hyperfine 
splitting increases or decreases on application 
of a field. With a field of 17.6 koe a shift of 
±2.65% is expected in line 6. 

The experimental technique was similar to that 
in our earlier work. 1 * 2 . 7 The carriage of a lathe 
was used to provide velocities by means of which 
the spectrum was scanned. The source was 
mounted in the narrow gap of an electromagnet 
capable of producing fields up to 20 koe. The 
magnet was attached rigidly to the end of the lathe 
and the absorber was mounted on the carriage. 
The result obtained for line 6 is shown in Fig. 2. 
On application of the field to the source, a shift 
to lower energy is unmistakable. The correlation 




10 II 

SPEED (mm/sec) 

FIG. 2. Line 6 observed with# ext = and # ext = 17.6 
koe, where # ex t is the external field applied to the 
source of resonance radiation . The ordinate is in units 
of 10 3 counts . 



514 



305 



Volume 4, Number 10 



PHYSICAL REVIEW LETTERS 



May 15, 1960 



is therefore negative. The magnitude of the ob- 
served shift is (2.7±0.4)% which is compatible 
with the linear relation H n =H no -H e xt- A negative 
shift of about the correct magnitude was also ob- 
served in line 4. The multiplet structure in line 
4 is symmetrical (Fig. 1) and so does not seriously 
interfere with the observation of a shift of its 
central member. 

The effective field at the iron nucleus has now 
been determined both in sign and magnitude. The 
existence of such a large negative field (-333 koe) 
was unexpected. Marshall 8 has discussed a num- 
ber of sources of the effective nuclear field. These 
consist mainly of direct effects of the 3d electrons 
and indirect effects of polarization of the various 
s electrons, which then contribute to the field via 
the Fermi contact interaction. The polarization 
of inner shells of electrons results in negative 
contributions to the field. In view of the experi- 
mental result these negative terms must completely 
dominate the other contributions. 

We are grateful to S. Raboy for loan of the mag- 
net . We wish also to thank M. R. Perlow for pre- 
paration of the source; F. J. Karasek for contin- 
uing to supply us with thin rolled iron foils; and 



E. Kowalski for assistance in taking the data. We 
have profited from a stimulating discussion with 
W. Marshall. 



This work was performed under the auspices of the 
U. S. Atomic Energy Commission. 

J G. J. Perlow, S. S. Hanna, M. Hamermesh, C. 
Littlejohn, D. H. Vincent, R. S. Preston, and J. 
Heberle, Phys. Rev. Letters 4, 74 (1960). 

2 S. S. Hanna, J. Heberle, C. Littlejohn, G. J. 
Perlow, R. S. Preston, andD. H. Vincent, Phys. Rev. 
Letters 4, 177 (1960). 

3 R. L. Mdssbauer, Z. Physik 151, 124 (1958). 

4 (Ferrocyanide) S. L. Ruby, L. M. Epstein, and 
K. H. Sun (to be published); (ferrocyanide, stainless 
steel) G. K. Wertheim, Phys. Rev. Letters 4, 403 
(1960); (Fe 2 3 ) O. C. Kistner and A. W. Sunyar, Phys. 
Rev. Letters 4, 412 (1960). 

5 A. C. Gossard, A.M. Portis, and W. J. Sandle 
(to be published) . 

6 G. T. Ewan, R. L. Graham, and J. S. Geiger (to 
be published) . 

7 S. S. Hanna, J. Heberle, C. Littlejohn, G. J. 
Perlow, R. S. Preston, andD. H. Vincent, Phys. Rev. 
Letters 4, 28 (1960). 

8 W. Marshall, Phys. Rev. 110, 1280 (1958), and 
private communication. 



306 

a^a a^a a^fa a^a a^a a^a a^a a^a a^a Jm M Ml <^fc- W a^a a^U a^a «*Jfc- a^a a^a Ji a^a a^> a^a a^a a^a a^e a^a a^a a^a a^a a^a a^a a^a a^a *^a a^a a^a a^a a^a a^a a^a a^a a^a ^i^l J| Ji^i^ll^l J|J| AJlU 

Volume 5, Number 8 PHYSICAL REVIEW LETTERS October 15, I960 



TEMPERATURE DEPENDENCE OF THE INTERNAL FIELD IN FERROMAGNETS* 

D. E. Nagle, H. Frauenf elder, + R. D. Tayloi, D. R. F. Cochran, and B. T. Matthias* 

Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 

(Received September 22, 1960) 



The saturation magnetization of a ferromagnet 
varies with temperature in a characteristic and 
well-investigated manner; it reaches a limiting 
value at temperatures well below the Curie tem- 
perature Tq and vanishes at and above Tq. The 
behavior of the internal magnetic field, however, 
is much less well known. Because the theories 
of internal fields in ferromagnets are still far 
from satisfactory, 1 accurate and detailed meas- 
urements of these internal fields over a wide 
temperature range are important; the effects 
near the Curie point are particularly revealing. 
Preliminary work on such problems has been 
discussed by the Argonne group. 1 In the present 
Letter, we report a determination of the temper- 
ature variation of the internal magnetic field in 
Fe (T c = 1043°K) and in a CoPd alloy (T c = 275°K) 
by means of the Mossbauer effect. 

That the MSssbauer effect is well suited for the 
investigation of internal fields no longer needs 
any justification. 1 The radioisotope Fe 57 is very 
convenient for such experiments, since Fe itself 
is a ferromagnet and since the MOssbauer spec- 
trum of its 14.4-kev gamma ray is well known. 2 
The emission spectrum of an Fe source at room 
temperature consists of six lines, well separated 
by the Zeeman effect due to the internal magnetic 
field. With an Fe absorber of identical internal 
field, these lines give rise to a MSssbauer spec- 
trum consisting of a prominent central absorp- 
tion line and five strong satellites on either side, 
two of them doublets. The internal field has been 
deduced from the splitting of these lines. 2 

Complications arise when the source and the 
absorber have different internal fields. The six 
emission and six absorption lines of different 
spacing then give rise to as many as 36 lines; 
the MOssbauer spectrum becomes harder to find 
and harder to identify. There are three ways to 
circumvent this difficulty. One can employ a 
source having no effective internal field where 
the six emission lines are collapsed into one, 3 
one can utilize a single line absorber, 4 ' 5 or one 
can reduce the number of lines by selecting 
plane 2 or circularly 6 polarized gamma rays. The 
best method will depend on the particular prob- 
lem. We have chosen the first approach for Fe 
and the third one, with selection of circularly 
polarized gamma rays, for CoPd. 



A CoPd source was prepared by electroplating 
Co" onto a CoPd alloy (92% Pd, 8% Co) and 
heating the CoPd foil in a vacuum furnace at 
1000°C for two hours. The source was then 
placed in a cryostat which allowed the source 
temperature to be varied from 88 C K to room tem- 
perature. The MSssbauer spectrum was observed 
with an Fe absorber, enriched to 75% Fe 57 and of 
equivalent thickness 2.2 mg/cm 2 . This absorber 
was mounted on a Jensen 8 -inch Flexair woofer 
and moved sinusoidally at a frequency of 11 sec" 1 . 
The output pulses from a scintillation counter 
were energy selected by a single-channel analy- 
zer and modulated with a saw tooth voltage, which 
was locked in with the speaker drive. The mod- 
ulated pulses were displayed on a 400-channel 
RIDL analyzer and thus yielded directly the de- 
sired MOssbauer spectrum, 4 as shown in Fig. 1. 
Due to the sinusoidal drive and linear display, 
the velocity scale in Fig. 1 is sinusoidal. The 
slight drop to the right in each spectrum is 
caused by dead-time effect in the 400-channel 
analyzer. 

A series of measurements was taken in which 
the temperature of the CoPd source was varied 
and the temperature of the Fe absorber was 24°C. 





•**■**.- - ( _, 5xlO S COUNTS /CHANNEL 




- -^^ *\ /"■* W ^"' V '*""*"*"**" W **—- »773*C 




v " Wsv " l "^.»» < , "**' 




♦—*•-. \ /'~~ *" ~"' % ~ 753#C 


UJ 




z 
z 
< 


**"*-H«V, ^»^-^/"^ S 738 * C 


X 

o 


•-— ^.^ +S\/**\>\j+^'** m *~ , ~**~ 722«C 


•^ 


— «. 


(f) 


». r\. '. 


z 


.- — •*•" v v-Saf^/^y'* ■•«■"•• — . 693 . c 


z> 




o 
o 






\ *""-> .—V - ^ 




■s v .v— . " 1MS •«' •..' \ / -•% W -~2I5*C 




r y rrs^v-- 



»5.32 mm /tec 

FIG. 1. Mossbauer spectra of a CoPd source at 
24 C C and an Fe absorber as functions of the absorber 
temperature. Positive velocity is taken to mean source 
moving away from absorber. 



364 



307 



Volume 5, Number 8 



PHYSICAL REVIEW LETTERS 



October 15, I960 



When the CoPd was above its Curie point of 
275°K, the six-line spectrum was obtained, sim- 
ilar to the lowest curve of Fig. 1; such a spec- 
trum is typical of a single-line source with an Fe 
absorber. Thus CoPd above its Curie point pos- 
sesses only a very small effective magnetic field: 
From the position of the lines and from the line 
widths, an upper limit of 2000 oersteds is ob- 
tained. Below the Curie point, the splitting pat- 
tern changes rapidly, indicating the appearance 
of an internal magnetic field. At 88°K, astonish- 
ingly enough, the pattern is typical of source and 
absorber with identical fields. Hence at 88°K, 
the field at the Fe 57 nucleus in CoPd is 3.3 xlO 5 
oersteds, 2 the same as that of Fe 57 in Fe at tem- 
peratures well below the Curie point. Details of 
these measurements will be published elsewhere. 7 

The single emission line of the CoPd source 
above its Curie point now offers a convenient tool 
for the investigation of the internal field in Fe. 
For this experiment, an absorber (5 mg/cm 2 Fe 57 , 
75% enriched) was placed in a furnace equipped 
with thin entrance and exit windows and contain- 
ing an atmosphere of hydrogen. The CoPd source 
was mounted on the speaker and the transmission 
spectrum recorded as a function of the absorber 
temperature. Some typical spectra obtained in 
this way are shown in Fig. 1. 

The curves in Fig. 1 show the decrease in the 
internal magnetic field, the temperature shift 8 ' 9 
of the center of the spectrum, the decrease in 
Mo'ssbauer absorption due to the Debye -Waller 
factor, and finally the disappearance of the effec- 
tive magnetic field in the Fe absorber at the Curie 
point. 

The relative magnetic field at the Fe 57 nucleus, 
as deduced from the curves in Fig. 1 and some 
additional data, is plotted in Fig. 2 as a function 
of T/Tq. For comparison, the relative satura- 
tion magnetization 10 is indicated by the solid line. 

We thank Dr. R. M. Bozorth for determining 
the Curie point of our CoPd alloy, Dr. C. E. 
Olsen for the preparation of the CoPd alloy and 
the annealing of the source, and Dr. W. E. Keller 
for the loan of his DYNA amplifier. We are grate- 
ful to Dr. P. P. Craig and Dr. J. G. Dash for 
stimulating discussions and to Mr. R. Hanft for 
his tireless efforts during the experiment. 





1 


i i i 


T 


1 ' 


! 1 1 


1.0 


: ■ 








- 








I> 


^V 


- 


03 










\l 






» RUN 1 








0.6 




• RUN 2 








0.4 










\ - 


0.2 




1 ! 1 


1 




X- 



FIG. 2. Relative internal magnetic field H(T)/H(2$TK) 
at the Fe 67 nuclei in an Fe absorber, as deduced by 
Mossbauer effect. The solid line indicates the relative 
saturation magnetization, cr(T)/o-(297°K), of Fe. 



*Work done under the auspices of the U. S. Atomic 
Energy Commission. 

tConsultant, University of Illinois, Urbana, Illinois. 

% Consultant, Bell Telephone Laboratories, Murray 
Hill, New Jersey. 

' Mossbauer Effect, Allerton House Conference, 
edited by H. Frauenfelder and H. Lustig (University 
of Illinois, Urbana, 1960). 

2 S. S. Hanna, J. Heberle, C. Littlejohn, G. J. 
Perlow, R. S. Preston, and D. H. Vincent, Phys. Rev. 
Letters 4_, 177 (1960). 

3 0. C. Kistner and A. W. Sunyar, Phys. Rev. 
Letters 4, 412 (1960). 

4 S. L. Ruby, L. M. Epstein, and K. H. Sun, Rev. 
Sci. Instr. 31, 580 (1960). 

5 G. K. Wertheim, Phys. Rev. Letters 4, 403 (1960). 

6 H. Frauenfelder, D. E. Nagle, R. D. Taylor, 
D. R. F. Cochran, and W. M. Visscher (to be pub- 
lished). 

7 R. D. Taylor, D. E. Nagle, H. Frauenfelder, and 
D. R. F. Cochran (to be published). 

8 R. V. Pound and G. A. Rebka, Jr., Phys. Rev. 
Letters 4, 274 (1960). 

9 B. D. Josephson, Phys. Rev. Letters 4, 341 (1960). 

10 American Institute of Physics Handbook (McGraw- 
Hill Book Company, New York, 1957), p. 5-208. 



365 



308 

Volume 5, Number 12 PHYSICAL REVIEW LETTERS December 15, 1960 



POLARIZATION OF THE CONDUCTION ELECTRONS IN THE FERROMAGNETIC METALS* 

A. J. F. Boyle, D. St. P. Bunbury, and C. Edwards 

The Physical Laboratories, University of Manchester, Manchester, England 

(Received November 15, 1960) 



It has recently been proposed 1 that the polar- 
ization of the 4s conduction electrons in Fe, Co, 
and Ni is antiparallel to that of the electrons in 
the unfilled 3d shell; this is the reverse of the 
usual assumption. There are few ways in which 
this polarization is manifested. One, however, 
is in the effective magnetic field at a nucleus, 
which acts through the contact interaction with 
the 4s electrons. The magnitude of this field has 
been given by Marshall 2 as 



H c = (8n/3)i s \ l p(0)\ A ' 



\i.np, 



(1) 



where \i is the Bohr magneton, £ s I ip{0) l^ 2 is 
the average probability density of a 4s conduc- 
tion electron evaluated at the nucleus [I^O)!^ 2 
being the free atom value], n is the number of 
conduction electrons per atom, and/) is their 
polarization. The sign of the field is positive, 
i.e., parallel to the direction of magnetization, 
if the polarization of the 4s electrons is parallel 
to the 3d polarization. 



Hanna et aL 3 > 4 have measured the magnitude 
and sign of the field at the Fe nucleus in iron by 
observing the Zeeman splitting of the 14-kev 
transition in Fe 57 and obtain the value -3 xlO 5 koe. 
In this case, however, there are other contri- 
butions to the field beside H c ; mainly those due 
to the electrons of the same atom, i.e., the 
polarized 3d electrons. These other contribu- 
tions almost certainly outweigh the effect of the 
field H c alone (H c is probably of the order of 50 
to 100 koe), and it is therefore difficult to draw 
any conclusion concerning its sign. 

In an attempt to measure H c directly, dilute 
(1%) solid solutions of Sn in the ferromagnetic 
metals (Fe, Co, and Ni) have been prepared. 
Since Sn is basically diamagnetic, we expect 
that there will be no contribution to the field at 
the Sn nucleus from its own inner electrons and 
that the field will be given simply by 



H =(4tt/3)M+// , 
s c 



(2) 
553 



309 



Volume 5, Number 12 



PHYSICAL REVIEW LETTERS 



Decembei 



I960 



where l^(0)l^ 2 in (1) is the appropriate value for 
a Sn atom. Since H c is due to an over -all polar- 
ization of the conduction electrons, the other 
terms should remain those appropriate to the 
solvent atoms. 5 

The Zeeman splitting of the 24-kev transition 
in Sn 119 was observed 8 using the MCssbauer 
effect. 7 ' 8 The transmission through the absorber 
(the alloy) of the radiation emitted by a moving 
source was measured as a function of the velocity 
of the source. The velocity spectrometer has 
been described in detail elsewhere. 6 The tem- 
perature of both the source and absorber were 
maintained around 100°K. Figures 1, 2, and 3 
show the transmission spectra obtained for Fe 
and Ni and Co. Each member of the doublet is 
an unresolved triplet caused by the splitting of 
the excited \ state, while the doublet separa- 
tion is due mainly to the splitting of the ground 
£ + state. The field strengths H s listed in Table I 
were derived using the known values of the mag- 
netic moments. (The excited state moment has 
been measured as 0.83 ±0.03 nuclear magneton. 9 ) 

The shifts listed in the table refer to the dis- 
placement of the spectra towards negative vel- 
ocity and are almost entirely due to the chemical 



Table I. Results derived from the velocity spectra 
of the absorption of 24-kev y rays from Sn 119m in ab- 
sorbers containing 1 % of Sn 119 dissolved in Fe , Co, 
and Ni. The shift is the displacement of the spectra 
towards negative velocity due to chemical effects, H s 
is the field at the Sn nucleus, and (4ti/3)M is the usuaJ 
Lorentz field. 



Solvent 


Shift 

(mm/sec) 


(koe) 




(4tt/3)M 


H s - (4tt/3)M 


Fe 


1.1 


-81 ±4 




7.5 


-88 


Co 


1.1 


-20.5 ±1 


5 


6.1 


-26.5 ±1.5 


Ni 


1.1 


+ 18.5±1 




2.0 


+ 16.5 ±1 



shift. 10 In this case, where the source was 
metallic tin, the magnitude of the shift should be 
proportional to (| s - £ Sn ) 10(0) l Sn 2 ; £ Sn and £ s 
refer to metallic tin and the solvent metal, 
respectively. 

The sign of H g was determined by placing each 
absorber in a transverse magnetic field of about 
7 koe; the magnetization direction is now parallel 
to the external field. A positive field was ob- 
served only in the case of Ni. The final values 




190 — 



188 



o magnetized, zero field 
• transverse field, 7 koe 

J I I I I L 



FIG. 1. The absorption 
spectra obtained with 1% Sn 119 
in Fe. Left scale, o ; right 
scale, • . 



94 



554 



310 

Volume 5, Number 12 PHYSICAL REVIEW LETTERS December 15, 1960 



velocity mm/sec 



FIG. 2. The absorption c 

spectra obtained with 1% Sn 119 o 

in Ni. Left scale, o ; right 
scale, • . 





-3 


-2 


-1 


O 


1 2 3 




1 


1 


1 


1 


1 1 1 • 












• _ 


40 






• 








• On 




• • - 




cP«* 




• ^ 




• 






39 


"% \ 




o 


• 






\\ 








T • 


38 


••X 


* 




< 


/ • 
P • 

• • 

» 


37 








o 

• 


• nickel 

unmagnetized 
transverse field, 6 koe 


36 


1 




1 


1 


1 1 1 



- 75 



- 73 



- 71 



69 



67 



-4 



40 



FIG. 3. The absorption 
spectrum obtained with 1 % 
Sn 119 in Co. 



39 — 



of [H s - (4tt/3)M] are listed in the table, where 
the values of M have been corrected for the 
quenching of the atomic moments by the added 
valence electrons of Sn. This is most significant 
in Ni, and, since it will occur predominantly 
amongst the nearest neighbors, will undoubtedly 
result in a lowered value of the observed field. 



velocity mm/sec 
2-1 O I 2 3 



1 


1 1 


1 1 1 


1 1 


1 

CD£Tc9 


— 








jff — 


o 

QiO 




<0 






1 


1 1 


unmagnetized 
1 1 1 


cobalt 

1 1 


1 



The results are obviously inconsistent with the 
existence of the single field H • firstly because 
of the different signs and secondly because of 
the relative magnitudes of the fields. Since p is 
very closely proportional to A/, 2 the expression 
(1) for H c can be written simply, H c = const £$M. 
Further, we can conclude from the chemical 






555 



311 



Volume 5, Number 12 



PHYSICAL REVIEW LETTERS 



December 15, I960 



shifts that | s is practically constant for Fe, Co, 
and Ni, and therefore H c will be roughly pro- 
portional to M. 

There is another mechanism 11 by which an 
effective field might be produced at the Sn nucleus 
in this situation. The wave function for the 4s 
electrons of the Sn atom will overlap with those 
of the 3d electrons of the surrounding solvent 
atoms, and the polarization of the latter will re- 
sult in a change in the relative spatial distribu- 
tion of the 4s electrons in respect of their spin 
orientation. In the region of the Sn nucleus, 
electrons with spin antiparallel to the 3d polar- 
ization will predominate, producing a negative 
effective field Hp. 

Evidence for the existence of such a field is 
provided by the results for Co (Fig. 3) which 
indicate the presence of a second field of -50 koe 
with the same chemical shift. X-ray analysis of 
the sample showed that both cubic and hexagonal 
structures were present in the rough proportion 
40:60. Since neither £ s nor p depends on the 
structure, the two values of the field cannot be 
associated with H c ; however, since the distri- 
bution of the 3d orbitals will probably differ for 
the two structures, 12 Hp would also differ. 

We have no estimate of Hp, but its magnitude 
should depend not only on M but also on the mean 
radius of the 3d shell and thus will decrease 
more rapidly from Fe to Ni than does the field 
H c which depends only on M. In the absence of 
any further contributions, combination of such a 
field Hp with a positive value of H c would there- 
fore provide a qualitative interpretation of the 
present results. 

It is interesting to compare the present results 
for the field at a Sn nucleus in Fe with those of 
Samoilov et al. 13 From measurements of the 
nuclear polarization these authors obtain values 



of 250 koe and 280 koe, respectively, for the 
field at In 114 and Sb 122 dissolved in Fe. 

We are indebted to Dr. C. Johnson for drawing 
our attention to the results of Samoilov et al. 
and we are grateful to many people for valuable 
discussion, particularly Dr. Lomer, Dr. W. 
Marshall, Dr. S. F. Edwards, and Dr. H. E. 
Hall. Much helpful advice in the preparation of 
the alloys has been given by Dr. J. Stubbles. 
We would also like to thank Dr. J. Zussman for 
performing the x-ray analysis. 



*Supported financially by the Department of Scientific 
and Industrial Research. 

'For a review article see C. Herring, Suppl. J. Appl. 
Phys. 31, 3S (1960). 

2 W. Marshall, Phys. Rev. 110, 1280(1958). 

3 S. S. Hanna, J. Heberle, C. Littlejohn, G. J. Per- 
low, R. S. Preston, and D. H. Vincent, Phys. Rev. 
Letters 4, 177 (1960). 

4 S. S. Hanna, J. Heberle, G. J. Perlow, R. S. 
Preston, and D. H. Vincent, Phys. Rev. Letters 4, 
513 (1960). 

5 D. R. Teeters, thesis, University of California, 
Berkeley, 1955 (unpublished). 

Separated Sn 119 used in the preparation of the alloys 
was supplied by the Chemistry Division, Atomic Energy 
Research Establishment, Harwell. 

7 R„ L. Mossbauer, Z. Physik 151, 125(1959). 

8 A. J. F. Boyle, D. St. P. Bunbury, C. Edwards, 
and H. E. Hall, Proc. Phys. Soc. (London) (to be pub- 
lished). 

9 A. J. F. Boyle, D. St. P. Bunbury, and C. Edwards 
(to be published). 

,0 O. C. Kistner and A. W. Sunyar, Phys. Rev. 
Letters 4, 412 (1960). 

n D. A. Goodings and V. Heine, Phys. Rev. Letters 
5, 370 (1960). 

12 J. B. Goodenough, Phys. Rev. 120, 67 (1960). 

13 V. N. Samoilov, V. V. Skliarevsky, and E. P. 
Stepanov, Soviet Phys. -JETP 11(38) , 261 (1960). 



556 



312 



Volume 6, Number 9 



PHYSICAL REVIEW LETTERS 



May 1, 1961 



HYPERFINE FIELD AND ATOMIC MOMENT OF IRON IN FERROMAGNETIC ALLOYS 

C. E. Johnson, M. S. Ridout, T. E. Cranshaw, and P. E. Madsen 

Atomic Energy Research Establishment, Harwell, England 
(Received February 1, 1961; revised manuscript received April 10, 1961) 



Mott and Stevens 1 and Lomer and Marshall 2 
have proposed models of the ferromagnetic alloys 
of the iron group metals, based on the assumption 
that for dilute alloys a rearrangement of electrons 
occurs around solute atoms only, while to a first 
approximation the electronic structure of the ma- 
trix atoms remains unaltered from that in the 
pure metal. Each atom carries a localized mag- 
netic moment which contributes directly to the 
total saturation moment and to the magnetic part 
of the neutron scattering cross section, and in- 
directly to the hyperfine field H n . Marshall 3 has 
shown that the hyperfine field in a pure ferromag- 
netic metal should be proportional to the magne- 
tization. This has been confirmed experimentally 
for cobalt 4 and for iron 5 by varying the tempera- 
ture, but the absolute agreement between theory 
and measurement is poor. 6 The Mo'ssbauer effect 7 
provides a method for measuring H n for iron in 
alloys which may be used to test the localized 
models and to investigate the relation between 
hyperfine field and atomic moment. 

We have measured the Mo'ssbauer absorption 
spectrum at room temperature over the whole 
range of Fe-Co and Fe-Ni alloys for the 14.4- 
key y radiation of Fe 57 . The source was pre- 
pared by electroplating Co 57 onto copper foil, 
followed by diffusion by annealing. Rapid elec- 
tron spin exchange resulted in a single line which 
had the natural width associated with the lifetime 
of the emitting state: The line was shifted by 0.20 
mm/sec with respect to the center of the iron 
spectrum. The alloys were prepared by arc- 
casting spectroscopically standardized materials 
supplied by Johnson Matthey, Ltd. , and were cold 
rolled into foils about 1 mil thick. Alloys contain- 
ing 30% or less iron were enriched in Fe 57 by 
plating and annealing. Data for pure cobalt and 
nickel were obtained with Co 57 sources plated 
onto foils of each metal, using stainless steel as 
a monoenergetic absorber. 8 Motion of the source 
was provided by a moving coil vibrator driven by 
an amplifier and a triangular wave generator. A 
moving iron transducer gave a voltage propor- 
tional to the source velocity, and this waveform 
was fed back to the input of the amplifier, so that 
the velocity of the source closely followed the in- 
put waveform. Counts were fed into a single- 



channel pulse-height analyzer to select the 14.4- 
kev radiation, and the output pulses were modu- 
lated with the velocity waveform and fed into a 
100-channel kicksorter. The resulting spectrum 9 
showed six lines arising from the Zeeman split- 
ting of the nuclear levels of Fe 57 , and the hyper- 
fine field was computed from their separations. 
The spectra for the alloys showed no appreciable 
line broadening or shifts compared with the pure 
iron spectrum. Hence, the variations in hyper- 
fine field due to local inhomogeneities are small 
(less than 3%), and there is no large change in 
s- electron density at the iron nuclei due to al- 
loying. 

The variation of the magnetic field at iron nu- 
clei in the alloys, expressed as a fraction of the 
field in metallic iron, is shown in Fig. 1, where 
H n {x)/H n {0) is plotted against the excess electron 
number x over that of iron. A remarkable feature 
is the general similarity in form with the corre- 
sponding region of the Slater -Pauling curve 10 for 
the saturation moments: For both alloy systems 
H n and the saturation moment show a maximum 
near x = 0.3. Even for small additions of solute 
it seems that large changes in the hyperfine field 
of the iron atoms result, in contrast to the local- 
ized theories. 

If it is assumed that the hyperfine field is pro- 
portional to the atomic moment in the alloys, then 
the moment on iron is given by fi(Fe) =2.22# w (x)/ 
H n (0) Bohr magnetons. The field on cobalt nuclei 
in Fe-Co alloys has been determined from low- 
temperature specific heat measurements by Arp, 
Edmonds, and Petersen, 11 and in contrast to the 
field on iron nuclei it shows no maximum but 
increases steadily from 217 xlO 3 gauss in pure 
cobalt to about 320 xlO 3 gauss in iron. From 
these data a curve for m(Co) may be derived, 
taking the moment in pure cobalt to be 1.71 Bohr 
magnetons. If these moments are averaged so 
that Ji =(l-c)u(Fe) + cju(Co), where c is the cobalt 
concentration, /i is found to lie on a curve which 
is close to the Slater -Pauling curve. 

There are no data on the hfs of nickel in Fe-Ni 
alloys, but Shull and Wilkinson 13 have measured 
the atomic moments in some of these alloys by 
neutron diffraction. For ordered Ni 3 Fe they find 
pt(Fe) =2.8, whereas our data, combined with the 



450 



313 



Volume 6, Number 9 



PHYSICAL REVIEW LETTERS 



May 1, 1961 



I.IO 


- r^°Q 






/ A 


o Fe - Co 




/ » 


• Fe-Ni 










\r A 




I.OO 




\ 


0.90 






0.80 







Co 



FIG. 1. The magnetic field at iron nuclei in Fe-Co 
and Fe-Ni alloys relative to the field in metallic iron, 
plotted as a function of electron number. The data in 
the range Fe 0=8 Ni 0o2 to Fe . 5 Ni 0#5 where the Curie 
points are low have been corrected to take account of 
incomplete saturation at room temperature. Points 
for Co and Ni agree well with the results given by 
Wertheim. 8 Alloys near Fe ,5Co . 5 are very brittle 
and difficult to roll and the points on the dashed curve 
were taken with small and cracked specimens. Alloys 
were also prepared in this range by electroplating and 
they gave higher values for H n which lie on the continu- 
ous curve. 



assumption that atomic moment is proportional 
to H n , give about 1.8. This implies that the pro- 
portionality between atomic moment and H n is 
not strictly valid in alloys. The discrepancy 
could be explained by a contribution to H n which 
depends on the nickel as well as the iron moments, 



e.g., if the component due to the conduction elec- 
tron polarization were a function of the average 
moment JL. Since p for NL,Fe is smaller than that 
for iron and# n is negative, 13 this explanation re- 
quires the polarization to be negative in accord 
with a suggestion of Anderson and Clogston 14 and 
with measurements of the field at tin nuclei in al- 
loys with iron. 15 An estimate of the effect of such 
a term in the Fe-Co alloys shows that, owing to 
the smaller variation of ju throughout the series, 
it would not destroy the agreement between /I 
derived from hfs and saturation magnetization 
data. 

We thank Dr. W. Marshall and Dr. W. M. Lomer 
for many valuable discussions, Dr. J. H. Stephen 
for performing the electroplating, and Dr. E. 
Bretscher for his generous support. 



*N. F. Mott and K. W. H. Stevens, Phil. Mag. 2, 
1364 (1957). 

2 W. M. Lomer and W. Marshall, Phil. Mag. 3, 185 
(1958) . 

3 W. Marshall, Phys. Rev. 110, 1280 (1958). 

4 A. M. Portis and A. C. Gossard, J. Appl. Phys. 
31, 205S (1960). 

5 D. E. Nagle, H. Frauenf elder , R. D. Taylor, 
D. R. F. Cochran, and B. T. Matthias, Phys. Rev. 
Letters 5, 364 (1960). 

8 S. S. Hanna, J. Heberle, G. J. Perlow, R. S. 
Preston, and D. H. Vincent, Phys. Rev. Letters 4, 
513 (1960). 

7 R. L. Mdssbauer, Z. Physik 151 , 124 (1958). 

8 G. K. Wertheim, Phys. Rev. Letters 4, 403 (1960). 

9 S. L. Ruby, L. M. Epstein, and K. H. Sun, Rev. 
Sci. Instr. 31, 580 (1960). 

10 See, e.g., C. Kittel, Introduction to Solid-State 
Physics (John Wiley & Sons, Inc. , New York, 1953), 
Chap. XII. 

U V. Arp, D. Edmonds, and R. Petersen, Phys. Rev. 
Letters 3, 212 (1960). 

12 C. G. Shull andM. K. Wilkinson, Phys. Rev. 97, 
304 (1955). 

13 In collaboration with Dr. G. J. Perlow we have shown 
that the sign of the field is negative for Fe 02 Ni 0#8 as 
well as for iron. 

14 P. W. Anderson and A. M. Clogston, Bull. Am. 
Phys. Soc. 6, 124 (1961). 

16 A. J. F. Boyle, D. St. P. Bunbury, andC. Edwards, 
Phys. Rev. Letters 5, 553 (1960). 



451 



314 



PHYSICAL REVIEW 



VOLUME 122. NU: 



Internal Magnetic Fields in Manganese-Tin Alloys* 

Luise Meyer-Schutzmeister, R. S. Preston, and S. S. Hanna 

Argonne National Laboratory, Argonne, Illinois 

(Received February 13, 1961) 

The hyperfine fields at the tin sites in two manganese-tin alloys have been studied as a function of tem- 
perature to above the Curie points. In addition to the Zeeman splittings, observed and analyzed previously, 
a possible quadrupole interaction of about 27 Mc/sec is observed in Mn 2 Sn. In Mn 4 Sn the hyperfine field 
is small and negative, about —45 koe; in Mn2Sn it is large and positive, about +200 koe. As in the case 
of the pure ferromagnetic transition elements, it seems necessary to invoke a positive term associated with 
conduction-electron polarization and a negative one arising from core polarization to explain these results. 



THE ferromagnetic alloys of manganese and tin, 
which were used previously 1 in observing the 
Zeeman splitting of the nuclear levels of Sn 119 by means 
of resonant absorption, 2 have now been studied more 
extensively to determine the nature of the internal 
magnetic field at the tin nucleus. The magnitude of 
the field has been measured as a function of tempera- 
ture, and the measurements have been carried above the 
Curie point in order to observe possible quadrupole 
or other interactions in the absence of complications 
produced by magnetic splitting. In addition, the sign 
of the field in each alloy was established by observing 







Mn 4 Sn 


39 mg/cm 2 




1.00 


,l,il 




\ 


9^165° C 




0.99 






1 / /o-^e 




0.99 




\ 


y /l35° C 


— 


1.00 




-^ 1 


J r*- 








0.99 






f 130° C 


— 






"°5\ V 




0.99 
1.00 


- 


V 


Jz: 


- 




" \ 






0.99 


- 


X 


^/ oo ° c 


- 


1.00 




" "s 


^o tyo - 








0.99 


- 


V 


"\A°° c 


- 


0.98 


1 


1 1 1 


i i 


i 



-8-4 4 8 

SPEED (mm/sec) 

Fig. 1. Resonant absorption in Mn 4 Sn, with a metallic Sn 119 
source at 77°K, for various absorber temperatures above and 
below the Curie point. Absorber thickness is 39 mg/cm 2 . 



* Work performed under the auspices of the U. S. Atomic 
Energy Commission. 

1 S. S. Hanna, L. Meyer-Schutzmeister, R. S. Preston, and D. 
H. Vincent, Phys. Rev. 120, 2211 (1960), hereafter referred to as I. 

2 R. L. Mossbauer, Z. Physik 151, 124 (1958). 



the change in the hyperfine structure on application 
of a large external magnetic field, as in our earlier work 
with iron. 3 

Except as noted below, the experimental technique 
was the same as used in I. In Fig. 1 are shown measure- 
ments on Mn 4 Sn at several temperatures from room 
temperature to above the Curie point at about 150°C. 4 
For these observations the Mn 4 Sn absorbing sample 
was clamped in vacuum in a frame which was warmed 
by an electrically heated coil of tungsten wire. The 
temperature was measured by a thermocouple in con- 
tact with the absorber. The Sn 119 source, which emits 
an unsplit line, was maintained at the temperature of 
liquid nitrogen. It will be recalled from I that the basic 
resonant absorption spectrum for Mn 4 Sn (with unsplit 
Sn source) consists of a doublet, each member of which 
is an unresolved triplet. The doublet separation is 
approximately equal to the magnetic splitting of the 
ground state. In Fig. 1 we can see this splitting decrease 




Fig. 2. Temperature variation of the internal field at the tin 
nucleus in Mn 4 Sn. A Curie temperature T c of 423°K has been 
assumed. Data taken from Fig. 1. 



3 S. S. Hanna, J. Heberle, G. J. Perlow, R. S. Preston, and D. 
H. Vincent, Phys. Rev. Letters 4, 513 (1960). 

<H. H. Potter, Phil. Mag. 12, 261 (1931). The Curie tempera- 
ture is given as 178°C by Ochsenfeld (reference 5). We continue 
to call this alloy Mn 4 Sn in accordance with much of the literature. 
However, the structure is close-packed hexagonal of the NisSn 
type (DO19). Actually, single phase samples appear to have a 
composition intermediate between Mn 3 Sn and Mn 4 Sn. We are 
greatly indebted to M. V. Nevitt for this information. 



1717 



315 



1718 



MEYER-SCHUTZMEISTER, PRESTON, AND HANNA 



400 



o 

v> 750 



744 



732 




•20 



10 

SPEED (mm/sec) 



Fig. 3. Resonant absorption in Mn 2 Sn, with a metallic Sn 119 
source at 77°K, at absorber temperatures below and above the 
Curie point. At 370°K the absorber thickness is 54 mg/cm 2 ; at 
80°K it is 91 mg/cm 2 . 



and then vanish as the temperature is raised to the 
Curie point. At these relatively high temperatures the 
absorption is very small and high precision has not been 
achieved in the data. The temperature variation of 
the internal field, as obtained from these and similar 
measurements, is displayed in Fig. 2. The data are 
normalized to the point obtained in I at the temperature 
of liquid nitrogen. The solid curve in the figure is a 
theoretical Weiss curve which is in rather good qualita- 
tive agreement with the experimental points. The value 
of the field at 0°K is estimated to be roughly 45 koe. 
In Fig. 3 the absorption spectrum of Mn 2 Sn is shown 
at two temperatures, one well below and the other 
well above the Curie point at — 11°C. S At the lower 
temperature one observes the Zeeman spectrum an- 
alyzed in I. The value of the internal field at 0°K is 
estimated to be about 200 koe. Above the Curie point 

6 C. Guillaud, thesis, Strasbourg, 1943; quoted in R. M. 
Bozorth, Ferromagnetism (D. Van Nostrand Company, Inc., 
Princeton, New Jersey), p. 340; R. Ochsenfeld, Z. Metallkunde 
49, 472 (1958). This alloy is reported to have a structure of the 
filled NiAs type, W. Hume-Rothery and G. V. Raynor, The Struc- 
ture of Metals and Alloys (The Institute of Metals, London, 
England). 



the spectrum collapses into a strong central absorption. 
Actually there is possibly a doublet structure in this 
central line as shown on an expanded velocity scale 
in Fig. 4. This structure is not strongly temperature 
dependent and would correspond to a quadrupole 
splitting of about 10 -7 ev or 27 Mc/sec. 

To obtain the sign of the internal field in these alloys, 
the powder sample, deposited on beryllium (0.010 in. 
thick), was clamped between two pieces of Lucite (each 
Y6 in. thick) and mounted in the gap (j in.) of an elec- 
tromagnet capable of producing fields up to 20 koe. 
A 0.001-in. foil of Pd was also inserted to reduce the 
25-kev x ray from tin. The magnetic splittings in the 
absorption spectrum were then compared with the field 
off and on. Because of the small aperture provided by 
the absorber in the gap it was desirable in obtaining 
the absorption spectra to have the source oscillate as 
close as possible to the absorber in order to increase 
the counting rate. Since it was essential to keep the 
source cold, it was mounted in vacuum on a horizontal 
copper bar attached to a horizontal reservoir of liquid 
nitrogen. With this arrangement it was possible to bring 
the source to within about 1 in. of the absorber mounted 
vertically in the gap. The value of the fringing field at 
the average position of the source was about one sixth 
the field in the gap. 

In the case of Mn 4 Sn the measurements could be 
carried out with the absorber at room temperature, 
at which temperature the internal field at the tin nucleus 
has a value of about 40 koe. 1 Relative to this field the 




-2 2 

SPEED (mm /sec) 

Fig. 4. Resonant absorption in Mn 2 Sn, with a metallic Sn 119 
source at 77°K, at two absorber temperatures above the Curie 
point. The velocity scale is expanded over that in Fig. 3. Absorber 
thickness is 54 mg/cm 2 . 



316 



INTERNAL MAGNETIC FIELDS IN MnSn ALLOYS 



1719 



applied external field of about 17 koe should produce 
an easily detectible shift. The observations are shown 
in Fig. 5. On application of the external field the doublet 
separation is seen to decrease. Hence the internal field 
at the nucleus in Mn 4 Sn is negative. The amount of 
the shift is compatible with the linear relation, 

Hn — HnQ — He^ty 

where H n and Hn are the hyperfine fields with and 
without the external field Z7 ex t- 

In the case of Mn 2 Sn it was necessary to cool the 
absorber below the Curie point (— 11°C) while keeping 
it in the gap of the electromagnet. To produce a con- 
venient internal field of about 40 koe the absorber was 
maintained at a temperature of about — 23 °C. This 
was accomplished by allowing a stream of nitrogen gas, 
cooled by passage through a coil immersed in liquid 
nitrogen, to strike each side of the Lucite holder con- 
taining the absorber. It was found that the temperature, 
measured with a thermocouple, could be held constant 
to within about 1° by carefully regulating the stream 
of cold nitrogen gas. It was necessary, however, to 
enclose the complete assembly (source, absorber, and 
pole pieces) in a plastic sheet in order to prevent mois- 
ture from condensing on the cold surfaces. The measure- 
ments were made by alternating many runs with field 
on and field off. The final averages are shown in Fig. 6. 
In this case the doublet separation is seen to increase 
on application of the field. Hence the hyperfine field 
at the tin nucleus in Mn 2 Sn is positive. The amount 



806- 



240 



w234 

O 




-4-2 2 4 
SPEED (mm/sec) 

Fig. 5. Resonant absorption in Mn 4 Sn at room temperature 
(~300°K) without an external magnetic field (above) and with 
an applied field of 17.5 koe (below). The splitting is decreased by 
application of the field. Absorber thickness is 42 mg/cm 2 . Source 
temperature is 77 °K. 



I ' I ' I ' I ' I ' I ' I ' I ' I 




676 



JLJL 



1.1,1,1,1 



-8-6-4-202468 
SPEED (mm/sec) 

Fig. 6. Resonant absorption in Mn 2 Sn at approximately 250°K 
without an external magnetic field (above) and with an applied 
field of 17.5 koe (below). The splitting is increased by application 
of the field. Absorber thickness is 45 mg/cm 2 . Source temperature 
is 77°K. 



of shift is compatible with the relation, 
H n =Hno-\-H eX f 

•Thus, a possible demagnetizing effect of the applied 
field in the sample is not noticeable in the above 
measurements. 

Since the hyperfine field is large and negative 
(~— 300 koe) in the pure ferromagnetic transition 
elements, 3,6 ' 7 it is of considerable interest to find such a 
large positive field (+200 koe) at the tin site in the case 
of Mn 2 Sn. Moreover, this field is very sensitive to the 
Mn : Sn ratio, since it changes sign in going to Mn 4 Sn. 
It is significant, perhaps, that the saturation magneti- 
zation is some four times as great in Mn 2 Sn as in Mi^Sn. 4 
The hyperfine field produced by the conduction electrons 
is given by Marshall 8 in the form 



ff.-(&r/3)»|*(0)|V, 



6 A. C. Gossard and A. M. Portis, Phys. Rev. Letters 3, 164 
(1959). 

7 S. S. Hanna, J. Heberle, C. Littlejohn, G. J. Perlow, R. S. 
Preston, and D. H. Vincent, Phys. Rev. Letters 4, 177 (1960). 

8 W. Marshall, Phys. Rev. 110, 1280 (1958). 



317 



1720 



ME YER-SCHt)TZMEISTER, PRESTON, AND HANNA 



where n is the number of conduction electrons per atom, 
|^(0) | 2 is the probability density of a conduction elec- 
tron at the nucleus, n is the Bohr magneton, and p is 
the polarization of the conduction electrons. It would 
appear that «|^(0)| 2 is roughly the same in the two 
alloys, since they show about the same isomer shift 9 
(see Figs. 1 and 3). 9a If, on the other hand, the polariza- 
tion of conduction electrons (produced by the adjacent 
magnetic electrons) increases with saturation magneti- 
zation, then the polarization and so also H c is greater 
numerically in Mn 2 Sn than in Mn 4 Sn. The simplest 
explanation of the large positive field in Mn 2 Sn is 
that H c is a positive field. 



9 O. C. Kistner and A. W. Sunyar, Phys. Rev. Letters 4, 412 
(1960). 

9s Note added in proof. We are indebted to V. Jaccarino for point- 
ing out that equality of the isomer shifts does not necessarily 
insure the above argument, since the polarization effect is localized 
at the top of the conduction band. 



As for the pure transition elements, 3 ' 10 ' 11 it is neces- 
sary to postulate, in addition to H c , the presence of a 
negative field, presumably associated with polarization 
of the core electrons of tin. In Mn 4 Sn this field pre- 
dominates. That this field is smaller, if different at all, 
in Mn 2 Sn is consistent with the fact that the Weiss 
field is smaller, since the Curie temperature is lower 
and the magnetization larger in Mn 2 Sn than in Mn 4 Sn. 

These effects observed in the manganese-tin alloys 
are somewhat analogous to those obtained by Boyle 
et al. n in dilute solutions of tin in the ferromagnetic 
transition elements. It is gratifying that essentially the 
same mechanisms 12 can be invoked to explain qualita- 
tively the observations on all these tin alloys. 

10 D. A. Goodings and V. Heine, Phys. Rev. Letters 5, 370 
(1960). 

11 A. J. Freeman and R. E. Watson, Phys. Rev. Letters 5, 498 
(1960). 

12 A. J. F. Boyle, D. St. P. Bunbury, and C. Edwards, Phys. 
Rev. Letters 5, 553 (1960). 



318 






PHYSICAL REVIEW- 



VOLUME T2 2, NUMBER 3 



Study of the Internal Fields Acting on Iron Nuclei in Iron Garnets, Using the Recoil-Free 
Absorption in Fe 67 of the 14.4-kev Gamma Radiation from Fe 57m f 

R. Bauminger, S. G. Cohen, A. Marinov, and S. Ofer 

Department of Physics, The Hebrew University, Jerusalem, Israel 

(Received December 22, 1960) 

The shape of the recoil-free absorption spectrum obtained in iron garnet absorbers has been investigated, 
using, as a source, a Co 57 source embedded in stainless steel. The results confirm the existence of two iron 
sublattices each showing a Zeeman structure characterized by different parameters. No significant differ- 
ences have been detected between the Zeeman structure in yttrium iron garnet and dysprosium iron garnet. 
The values obtained for the effective magnetic field at the Fe 57 nuclei at room temperature are 3.90X10 5 oe 
and 4.85 X10 5 oe for the d and a iron lattice sites, respectively. At liquid air temperature the corresponding 
fields are 4.6X10 5 oe and 5.4X10 5 oe, respectively. The mean value of the chemical shift for the d sites rela- 
tive to stainless steel is about 0.04±0.005 cm/sec and about 0.06±0.005 cm/sec for the a sites. 



INTRODUCTION 

THE phenomenon of recoil-free resonance absorp- 
tion (Mossbauer effect) 1 of nuclear gamma rays 
has already been shown to constitute a powerful tool 
for investigating the properties of the internal fields 
acting on nuclei in solids. In particular, the 14.4-kev 
gamma rays of Fe 57m have been employed in a striking 
way to study the internal fields at iron nuclei in ferro 
and antiferromagnetic materials. 2-4 

In these cases the Zeeman splitting patterns obtained 
in the absorption spectra can be simply interpreted. 5 
From a knowledge of the nuclear magnetic moment of 
the nuclear ground state, values for the effective mag- 
netic fields at the iron nucleus can be obtained and 
also in some cases the magnitude of the quadrupole 
interactions. 

In the magnetic materials investigated so far, the 
iron atoms occupy equivalent lattice positions and the 
local fields at the iron nuclei are characterized by a 
single set of parameters. In the present investigation, 
a study has been made of the local fields in a ferri- 
magnetic material in which the iron atoms occupy 
two nonequivalent sets of positions. It was thought 
that this might result in a difference in the effective 
magnetic fields at the iron nuclei in the two sites. The 
ferrimagnetic materials studied were the iron garnets 
(stoichiometric formula 5 Fe203-3M 2 3 , where M in- 
dicates a rare earth ion or yttrium). The magnetic 
properties of these materials have been studied inten- 
sively in recent years. 6 Pauthenet has shown that in 
order to explain these properties it is necessary to 
assume that the two iron sublattices in this structure 
have opposite and unequal magnetizations. The garnets 



f Supported in part by the U. S. Air Force, Air Research and 
Development Command through its European Office. 

1 R. L. Mossbauer, Z. Phvsik, 151, 124 (1958). 

2 S. S. Hanna, J. Heberle, C. Littlejohn, G. J. Perlow, R. S. 
Preston, and D. H. Vincent, Phys. Rev. Letters 4, 177 (1960). 

3 G. K. Wertheim, Phys. Rev. Letters 4, 403 (1960). 

4 0. C. Kistner and A. W. Sunyar, Phys. Rev. Letters 4, 412 
(1960). 

6 G. DePasquali, H. Frauenfelder, S. Margulies, and R. N. 
Peacock, Phys. Rev. Letters 4, 71 (1960). 

6 R. Pauthenet, Ann. Phys. 3, 424 (1958). 



have a body-centered cubic structure, space group Oh 10 , 
the unit cell containing 96 oxygen ions in the general 
lattice positions, with 16 Fe 3+ ions in the a special 
positions (octahedral sites) and 24 Fe 3+ ions situated 
in the d special positions (tetrahedral sites). These 
constitute the two iron sublattices. The yttrium or 
rare earth ions occupy the 24 c dodecahedral sites. 
The unit cell edge is about 12 A. 

Two representative garnets of the above type were 
studied, yttrium iron garnet and dysprosium iron 
garnet. The yttrium ion is diamagnetic, whereas the 
dysprosium ion is strongly paramagnetic. It is known 
that in these garnets there is an exchange field which 
tends to align the rare earth ions relative to the mag- 
netization of the iron ions. Although this exchange 
interaction is known to be small relative to the domi- 
nant exchange interaction, which occurs between the 
two iron sublattices (10 5 oe compared to 6X10 6 oe), 7 
it was thought of interest to see whether this could 
give rise to a difference in the effective field at corre- 
sponding iron nuclei for the yttrium and dysprosium 
garnets, respectively, at a given temperature. 

Since the magnetization of the garnets is a function 
of temperature 6 it was also considered interesting to 
try to correlate the effective fields at the nucleus with 
temperature. 

EXPERIMENTAL 

In these experiments the absorption of the 14.4-kev 
gamma ray of Fe 57m bound in stainless steel was meas- 
ured in polycrystalline garnet absorbers, containing 
iron enriched in Fe 57 , as a function of the relative 
velocity between source and absorber. A stainless steel 
source containing Co 57 has already been shown to give 
a relatively narrow unsplit emission line. 34 Such a 
source can be used very effectively to explore the ab- 
sorption spectrum of an absorber containing Fe 57 . A 
large argon-filled proportional counter was used to de- 
tect the 14.4-kev radiation and provided a better reso- 
lution of this radiation against background than a thin 



7 P.-G de Gennes, C. Kittel, and A. M. Portis, Phys. Rev. 
116, 323 (1959). 



743 



319 



744 



BAUMINGER, COHEN, MARINOV, AND OFER 




Fig. 1. The absorption by stainless steel of the 14.4-kev gamma 
ray emitted in the decay of Fe 67m embedded in stainless steel, as 
a function of relative velocity between source and absorber. 

Nal(Tl) scintillation counter. As absorbers, two types 
of garnets were used in this work, yttrium iron garnet 
and dysprosium iron garnet. These were synthesized 
from Y2O3 or Dy 2 03 and Fe203, containing iron en- 
riched in Fe 57 (70%) in order to maximize the Moss- 
bauer absorption relative to competing processes. X-ray 
photographs taken by Mr. Kallman of this laboratory 
showed a unique crystal structure for the samples. 
The samples were ground fine, spread out to an average 
thickness of about 10 mg/cm 2 and held rigidly be- 
tween thin Lucite disks, and in this form used as 
absorbers. 

At first, the Doppler shift between source and ab- 
sorber was provided by a mechanical device 8 consisting 
of a uniformly rotating eccentric wheel driving a 
carriage bearing the source, whose speed could be 
varied via a coupling. Later an instrument was de- 
veloped which enabled the counting rate to be auto- 
matically recorded as a function of the relative velocity, 9 
and the final results presented here were carried out in 
this way. The source was mounted on a loudspeaker 
membrane and vibrated sinusoidally at 65 cycles/sec. 
The amplitude of the output of the pulse-height se- 
lector, channelled on the 14.4-kev peak recorded in the 
proportional counter, was modulated in appropriate 
phase with the sinusoidal motion, so that the modu- 
lated part of the amplitude would be proportional to 
the source velocity at the time of emission of the corre- 
sponding photon. The spectrum of modulated pulses 
was displayed on the first 128 channels of a 256-channel 
pulse-height analyzer. As a consequence of the har- 

8 S. Ofer, P. Avivi, R. Bauminger, A. Marinov, and S. G. 
Cohen, Phys. Rev. 120, 406 (1960). 

•The instrument used was similar to a device constructed 
recently by Dr. E. Sunyar of Brookhaven National Laboratory. 
We are indebted to him for information concerning his instru- 
ment, particularly the electronic circuits. 



monic motion, the time the source spends in each 
velocity channel is itself a function of velocity. In order 
to obtain the true spectrum of counting .rate as a 
function of velocity, the spectrum recorded on the 
analyzer was normalized in the following way. The 
output pulses from an independent scintillation counter 
and radioactive source were treated in an identical 
fashion as described above for the pulses from the 
proportional counter and displayed on the second half 
of the multichannel analyzer. The output of the two 
counters were alternately switched every half-minute 
into the single-channel analyzer equipped with modu- 
lated output and the two spectra stored in the two 
halves of the analyzer. Identical counting rates were 
arranged in order to insure similar counting losses in 
the two halves of the multichannel analyzer. The 
normalized spectrum was obtained by dividing the 
number in a given channel of the first half of the 
analyzer by the number in the corresponding channel 
in the second half. This method of normalization en- 
sured good stability over long periods of time against 
drift in the electronic instrumentation. In practice, a 
scale of velocities was established by using as a calibra- 
tion the absorption spectrum of antiferromagnetic 
Fe 2 03 and relying on the velocity assignments to the 
peaks in the Zeeman pattern obtained by Kistner and 
Sunyar 4 and as confirmed, also, by measurements in 
this laboratory, using the mechanical device providing 
uniform relative velocity mentioned at the beginning 
of this section. In both the latter measurements the 
relative velocities were measured directly. A typical 
absorption spectrum of Fe 2 3 , using Co 57 embedded in 
stainless steel, is shown in Fig. 2. Figure 1 shows the 
results obtained using a stainless steel absorber with 
the same source and confirms that an unsplit emission 
line is indeed obtained. The width of the absorption 
line is about four times the natural linewidth. Assuming 
an emission and absorption line of the same shape in 
the stainless steel source and stainless steel absorber, 
respectively, one concludes that the width of the 
emission line is about twice the natural width. 

Measurements were carried out for both types of 
garnet absorbers at room temperature and also at the 
temperature of liquid air. The latter experiments were 
carried out with the garnet absorbers immersed in 
liquid air contained in a Styrofoam container. 

RESULTS AND DISCUSSION 

Figures 3 and 4 show the results obtained with the 
yttrium garnet absorber at room and liquid air tem- 
peratures, respectively. These spectra should be com- 
pared with the spectrum obtained with antiferromag- 
netic Fe 2 3 shown in Fig. 2. The latter spectrum has 
already been investigated by Kistner and Sunyar 4 and 
analyzed in terms of a Zeeman splitting of the nuclear 
levels produced by a single magnetic field at the iron 
nucleus (5.15X10 5 oe) somewhat modified by a quad- 



320 

INTERNAL FIELDS ACTING ON Fe NUCLEI IN Fe GARNETS 745 




8 mm/see 



Fig. 2. The absorption at room temperature by Fe 2 Oj of the 14.4-kev gamma ray emitted in the decay of Fe"" 
embedded in stainless steel, as a function of relative velocity between source and absorber. 



rupole interaction of the Fe 87 nucleus with a finite 
electric field gradient at the nucleus. 

It is seen that there is an essential difference between 
the garnet spectra and the Fe 2 03 spectrum, the garnet 
spectra consisting of a series of doublets, but otherwise 
resembling the Fe2C>3 spectrum. An actual splitting is not 
apparent in the lines close to zero velocity (i.e., for the 
two "3" and "4" lines in Fig. 3 and for the three "2", 
"3" and "4" lines in Fig. 4) but these lines are wider 
than would be expected assuming a single field. This 
pattern in the garnets is in fact exactly that to be ex- 
pected if the conjecture mentioned in the Introduction 
is correct, namely, that the iron nuclei situated in the 
two different lattice sites experience local fields which 
are appreciably different. One would then expect to 



obtain a superposition of two patterns, whose intensi- 
ties are in the ratio of 3:2 (ratio of iron atoms in the 
two nonequivalent sites) and each resembling in shape 
that of Fe 2 03 to a first approximation, but character- 
ized by spacings determined by different parameters. 
The spectra have been analyzed in accordance with 
this interpretation. 

In the cases when the doublets are resolved, the 
stronger component is assigned to the d sites (which 
are more numerous than the a sites in the ratio 3:2) 
and are labeled d in the figures, and the weaker com- 
ponent is assigned to the a sites. For the spectrum 
taken at room temperatures (Fig. 3) the positions of 
the four resolved lines "Id," "2d" "5d," and "6d" 
were used to calculate the four parameters determining 







o 














o 


1.00 




o 
















IS 

1 0.96 

z 

3 




9 °o\ 




A° 

01 


y°\ /• 
/I /° 


°\° 


o o 
6 \° 6° ° 

F °\ / 
°\° / 


a. / °\ 


°f\ / 


8 0.94 














\° / 






RELATIVE 




1 1 


10 


uV 


2 « la 

i i i 


3 

1 


o\ J 


sd 5a 

1 1 1 


j 6a 



-8 -7 



-5 



-3 -2-10 1 
VELOCITY 



8 mm/sec 



Fig. 3. The absorption at room temperature by yttrium iron garnet of the 14.4-kev gamma ray emitted in the decay of 
Fe 67m embedded in stainless steel, as a function of relative velocity between source and absorber. 



321 



746 



BA.UMINGER, COHEN, MARINOV, AND OFER 



1.00 


-a° 







o 


o ° 






,0 o 


o 
o o 


o 


• 


£ 








o o A o 




\ f a 










*7"^ <a f 


£0.98 


- 






°\ 


/ o o 


\ 7 




' o 






1 / 1 / 


pO.<J6 

z 






of 


\ °i 


po 


\ 7° 

°\ / 


10 H 

° V / 




1° 




\ / 6d 


3 
























0.94 












3 


4- 






sa 


°lol 


Ul 

>0.<?2 




10 




2 










5dV 




od 


< 






id 


















cJo.po 


i 


l 


r 


I I 


I I 


I I I 


1 


1 


1 


1 


1 1 1 



-6 



-2 



2 

VELOCITY 



8 mm/sec 



Fig. 4. The absorption at liquid air temperature by yttrium iron garnet of the 14.4-kev gamma ray emitted in the decay 
of Fe 57m embedded in stainless steel, as a function of relative velocity between source and absorber. 



the Zeeman pattern of the iron nuclei in the d sites, 
and the positions of the four corresponding a lines to 
calculate the parameters of the nuclei at the a sites. As 
described in the work of Kistner and Sunyar, 4 the 
Zeeman spectrum is characterized by four parameters 
and these are uniquely determined by the position of 
four peaks. The parameters are, in the notation of 
Kistner and Sunyar: g and gi, the magnetic splitting 
parameters for the \ and f nuclear levels, respectively ; 
AE, the shift between the center of gravity of the ab- 
sorption lines and the emission lines of Fe 67 in stainless 
steel; and e, the quadrupole interaction parameter, 
shifting only the substates belonging to the upper f 
level. When the parameters were found in this way, the 
expected positions of the lines "3a" and "3d," "4a" 
and "Ad" were calculated, and found to be in very 
good agreement with the position of the experimentally 
unresolved lines "3" and "4," thus demonstrating the 
consistency of the analysis. Moreover, the values of 
go/gi (equal to the ratio of the nuclear g factors in the 
J and f states) obtained are in satisfactory agreement 
with the value obtained by Hanna et al? and Kistner 
and Sunyar. 4 

In the analysis of the spectrum taken at liquid air 
temperature (Fig. 4) in which only three clearly re- 
solved doublets are seen, the four characteristic pa- 
rameters for each site were calculated, assuming a 



value of 1.77 for go/gi and using the position of the 
three resolved lines appropriate to each lattice site, 
i.e., "Id," "5d," and "6d" for the d sites, and "la," 
"5a," and "6a" for the a sites. As in the previous 
example, the expected positions of the remaining lines 
were calculated from these parameters thus obtained 
and found to be in good agreement with the position 
of the observed unresolved peaks "2," "3," and "4." 

The spectra obtained with the dysprosium garnet 
absorber are not shown since they so closely resemble 
those obtained with yttrium garnet at the same tem- 
perature. They were analyzed in a similar way. The 
final values of the characteristic parameters go, g\, AE, 
and e for the two sites in the various experiments are 
given in Tables I and II. Table I, showing the results 
obtained at room temperatures, also shows the values 
01 go/gi obtained directly from the experiments in each 
case. The value of the effective magnetic field at the 
nucleus, H e u, in each case was calculated from the 
value obtained for g using a value 10 of +(0.0903 
±0.0007) nm for the magnetic moment of the nuclear 
ground state of Fe 57 . The values of Z7 e ff, determined in 
this way, are given in Tables I and II. 

The results demonstrate that the a and d sites are 
indeed characterized by considerably different values 
of H e a. Thus the values obtained for H ef f at room tem- 
perature are 3.9X10 6 oe at the d sites and 4.85 X10 6 



Table I. Results of analysis of measurements on yttrium and dysprosium iron garnets at room temperature. 





Y 3 Fe 5 0i2 (yttrium iror 


garnet) at 300°K 


Dy 3 Fe 5 0i2 (dysprosium 


iron garnet) at 300 C K 




Strong spectrum 


Weak spectrum 


Strong spectrum 


Weak spectrum 




d sites 


a sites 


d sites 


a sites 


go (cm/sec) 


0.46±0.02 


0.58±0.02 


0.460±0.025 


0.580±0.025 


gi (cm/sec) 


0.270±0.015 


0.325±0.015 


0.275±0.020 


0.325±0.020 


AE (cm/sec) 


0.035±0.010 


0.055±0.010 


0.05±0.01 


0.06±0.01 


e (cm/sec) 


0.00±0.01 


0.00±0.01 


0.00±0.01 


0.00±0.01 


go/gi 


1.7±0.1 


1.8±0.1 


1.7±0.1 


1.75±0.1 


U M (oe) 


(3.90±0.1)X10 5 


(4.85±0.15)X10 6 


(3.95±0.15)X10 6 


(4.85±0.20)X10 6 













G. W. Ludwig and H. H. Woodbury, Phys. Rev. 117, 1286 (1960). 



322 

INTERNAL FIELDS ACTING ON Fe NUCLEI IN Fe GARNETS 747 

Table II. Results of analysis of measurements on yttrium and dysprosium garnets at liquid air temperature. 



Y 3 Fe 5 0i2 (yttrium iron garnet) at 85°K 
Strong spectrum Weak spectrum 



Dy 3 Fe 6 0i2 (dysprosium iron garnet) at 85°K 



d sites 



a sites 



Strong spectrum 
d sites 



Weak spectrum 
a sites 



go (cm/sec) 
g, (cm/sec) 
AE (cm/sec) 
« (cm/sec) 
#ett (oe) 



0.550±0.015 

0.31±0.01 

0.04±0.01 

0.00±0.01 
(4.60±0.15)X10 5 



0.635±0.015 
0.36±0.01 
0.06±0.01 
-0.010±0.01 
(5.35±0.15)X10 6 



0.545±0.020 

0.310±0.015 

0.030±0.015 

0.00±0.015 

(4.6±0.2)X10 6 



0.64±0.02 

0.360±0.015 

0.060±0.015 

-0.010±0.015 

(5.4±0.2)X10 6 



oe at the a sites. The results at liquid air temperature 
give higher values for H e u, but still different for the 
two sites — 4.6X10 5 oe and 5.35 oe for the d and a 
sites, respectively. 

Nagle et al. n have recently shown that in ferromag- 
netic iron, H e u, which has a well-defined value for 
temperatures below the Curie temperature, shows the 
same functional dependence on the temperature as the 
relative saturation magnetization M„ from tempera- 
tures at which the magnetization is almost saturated 
up to the Curie temperature. This remarkably simple 
result seems to demand that the fluctuations in the 
value of the nuclear field at a particular iron nucleus 
must take place in a time short compared to the 
Larmor precession period of the nucleus in the magnetic 
field produced by the extranuclear electrons. 

These fluctuations may be expected to follow the 
fluctuations of the magnetic moment of the extra- 
nuclear electrons. The temperature dependence of the 
expectation value of the extranuclear moment in the 
direction of magnetization will then determine the tem- 
perature dependence of both the microscopic H e a at 
the nucleus and the macroscopic magnetization. It is 
of interest to test these ideas in the case of the garnets 
and to see whether the values of H e a at different 
temperatures are proportional to the values of the 
partial spontaneous magnetizations of the appropriate 
sublattice. The partial magnetizations cannot of course 
be directly measured for ferrimagnetic materials. 
Pauthenet, however, starting from the experimental 
results for the relative saturation magnetization of the 
garnets as a function of temperature, and using the 
Neel two-sublattice model, 12 has calculated the partial 
spontaneous magnetization for yttrium ion garnet as 
a function of temperature for the two iron sublattices. 13 

Table III shows a comparison between the ratio of 
the values of H e a at 85 °K and 300°K in yttrium iron 
garnet for the a and d sites and the corresponding ratio 
of the spontaneous magnetization per ion (m) as calcu- 
lated by Pauthenet. 6 

The agreement is seen to be quite good and suggests 
that in this case, also, H e u follows the variation in 
magnetization. It should be pointed out that the calcu- 



lations of Pauthenet, based on the Neel model, indicate 
that at liquid air temperatures the partial magnetiza- 
tions of the a and d lattices should both be very close 
indeed to saturation, corresponding to a value of 5 hb 
per ferric ion. Nevertheless, the values of H e u remain 
different for the a and d sites. Assuming saturation 
really occurs at liquid air temperature, this behavior 
implies a difference in the extranuclear electronic con- 
figurations determining the nuclear field in the two sites. 

As seen from the results in Tables I and II, no 
significant difference in the values of H e a and the other 
parameters between the corresponding sites in yttrium 
and dysprosium garnets has been detected. The local 
fields at the corresponding sites in the two garnets 
cannot differ by more than a few percent. This is not 
unreasonable in view of the relatively small value of 
the exchange interaction acting between the rare earth 
ions and the ferric ions. 

It is not to be expected that a reliable value of the 
quadrupole interaction can be obtained from measure- 
ments on polycrystalline materials, since the shift in 
the sublevels due to the quadrupole interaction is a 
function of the angle between the magnetic field and 
the direction of the field gradients which are well 
defined with respect to the crystallographic axis. In 
fact, for the case of an axially symmetric field gradient 
and completely random orientation between the direc- 
tion of magnetic field and the field gradient, the average 
quadrupole shift, to first order, would be zero, and only 
a line broadening would be produced. In ferromagnetic 
and ferrimagnetic materials the correlation which in 
general will exist between the orientation of a crystallite 
and the orientations of the domains within this crystal- 
lite may very well lead to an average value of the 
quadrupole shift different from zero. The present re- 
sults show no evidence of an appreciable quadrupole 
Shift, but in view of the above, little can be deduced 



Table III. Comparison of ratio of value of H e u at 85°K and 
at 300°K, with ratio of partial magnetizations at these tempera- 
tures as calculated by Pauthenet, for yttrium iron garnet. 



d sites 



11 D. E. Nagle, H. Fraunfelder, R. D. Taylor, D. R. F. Cochran, 
and B. T. Matthias, Phys. Rev. Letters 5, 364 (1960). 

12 L. Neel, Ann. Phys. 3, 137 (1948). 

13 See reference 6, Fig. 12 and p. 454. 



tfeff(85 o K)/tf eff (300°K) 
w(85°K)/w(300°K) 



1.16 
1.17 



1.11 
1.08 



323 

748 BAUMINGER, COHEN, MARINOV, AND OFER 

concerning an upper limit for the quadrupole interac- isotope shift should certainly be influenced by changes 

tion at the iron nucleus. in the value of s wave functions at the nucleus and it is 

Concerning the chemical shifts, there is no evidence possible that H e!{ will also be influenced by these 

of any temperature dependence. The values of the changes. 

chemical shift however, do seem to be consistently ACKNOWLEDGMENTS 

greater at the a sites than the value at the d sites. The authors would like to thank M. Schieber of the 

Noting that H et( is greater at the a sites than at the department of electronics, the Weizmann Institute, for 

d sites, one might speculate on a possible correlation preparing the garnets, E. Sunyar of Brookhaven 

between chemical shift and the saturation value of National Laboratory for information concerning elec- 

H e n. If an appreciable part of the chemical shift is due tronic circuits, E. Segal for help in constructing the 

to an isotope shift, as has been suggested by Kistner apparatus, and A. Mustachi for help in chemical 

and Sunyar, 4 such a correlation may arise since the problems. 



324 



ON THE USE OF THE MOSSBAUER EFFECT 
FOR STUDYING LOCALIZED OSCILLATIONS 
OF A TOMS IN SOLIDS 

S. V. MALEEV 

Leningrad Physico- Technical Institute, 
Academy of Sciences, U.S.S.R. 

Submitted to JETP editor June 29, 1960 

J. Exptl. Theoret. Phys. (U.S.S.R.) 39, 891-892 
(September, 1960) 

1HE Mossbauer effect consists in the emission 
(or resonant absorption) by a nucleus in a solid 
of a y quantum with an energy which is precisely 
equal to the energy of the transition, because of 
the fact that the recoil momentum is transferred 
to the crystal as a whole. 

Usually the nucleus which radiates the y quan- 
tum is formed by the decay of some other nucleus. 
As a result of this process, the nucleus can with 
a very high probability leave its place in the lat- 
tice and get stuck somewhere at an interstitial 
position. But, even if the nucleus does not move 
about, if it should change its atomic number as 
a result of the decay the forces holding it in the 
lattice will change. Thus the nucleus emitting the 
Mossbauer quantum must be a lattice defect. 

On the other hand it is well known (cf. refer- 
ence 1) that the spectrum of oscillations of a de- 
fect atom in a lattice consists of a continuous 
spectrum, coinciding with the spectrum of oscil- 
lations of the ideal lattice, and of discrete fre- 
quencies which do not coincide with any of the 
frequencies of normal vibrations of the atoms of 
the ideal lattice. Vibrations with such frequen- 
cies ( localized oscillations ) cannot propagate 
through the lattice over any sizeable distance. 

At the same time there is a finite probability 
that in the emission of a y quantum there is si- 
multaneously emitted or absorbed (the latter, 
naturally, only for sufficiently high temperatures, 
T Z Ho)l, where u>l is the frequency of the local- 



ized oscillation) a quantum of the localized oscil- 
lation. Thus, the spectrum of emitted y quanta 
will consist of an unshifted line corresponding to 
the energy of the transition and of a continuous 
background corresponding to the emission and 
absorption simultaneously with the y quantum of 
phonons from the continuous part of the spectrum 
of oscillations of the atom; on this background, 
there will be individual discrete peaks due to the 
emission and absorption of quanta of the localized 
oscillations. 

These peaks can be observed in almost the 
same way as the unshifted line is observed. 
Namely, an absorber containing atoms in the 
ground state should be moved with such a veloc- 
ity that the Doppler shift of its undisplaced ab- 
sorption line will be equal to the frequency of the 
localized oscillation. One then will observe a 
stronger absorption than for neighboring frequen- 
cies. The velocity needed for this is obviously 
determined by the condition col = vw/c, where 
u> is the frequency of the y line. If the energy 
of the transition is of the order of tens of kev, 
and hcoL ~ 0.01 ev, v ~ 10 3 — 10 4 cm/sec. Such 
a velocity is not difficult to obtain by placing the 
absorber on the rim of a rotating disk. 



^aradudin, Mazur, Montroll, and Weiss, Revs. 
Modern Phys. 30, 175 (1958). 

Translated by M. Hamermesh 



325 



Polarization of Co" in Fe Metal* 

J. G. DASH.t R. D. Taylor, D. E. Nagle, P. P. Craig, and W. M. Visscher 

Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 

(Received December 16, 1960) 

A study has been made of the effect of low temperatures on the resonant emission and absorption of 
14.4-kev Mossbauer radiation from Fe 57 in Fe metal. Analysis of intensity changes in the hyperfine spectrum 
is made in terms of the Zeeman level splittings of the ground states of Fe 57 absorbing nuclei and of the 
ground states of Co 57 parent nuclei. The theory for the temperature dependence is developed in terms of 
the properties of the Co 67 decay and of the subsequent gamma transitions. Experiments were carried out 
with a source of Co 67 nuclei in Fe metal at temperatures between 4.5° and 0.85°K. The experimental results, 
analyzed in terms of the theory, yield a value of the hyperfine magnetic field at the Co 67 nuclei. Compar- 
ison of the result with other pertinent experimental values indicates that depolarization of the nuclei by the 
/sT-capture decay of Co 67 is not evident in the present material. 



I. INTRODUCTION 

THE 14.4-kev gamma rays of Fe 67 nuclei are known 
to have, in suitable crystals, a high proportion / 
of recoil-free, or "Mossbauer" radiation. 1-4 Interest in 
the Fe 67 system is enhanced by the relatively narrow 
linewidths characteristic of the excited state (lifetime 
10~ 7 sec) and the clearly resolvable hyperfine com- 
ponents of the Mossbauer pattern. Experiments with 
Fe crystal sources and absorbers are facilitated by the 
large value of / even at room temperature as a conse- 
quence of the high Debye characteristic temperature 
(0~42O°K) and the low nuclear recoil temperature 
(rR^40°K). A source of Co 67 nuclei dissolved in a Fe 
lattice at room temperature has /c^O.7 1 ; cooling the 
source to 0°K increases / to 0.92. This limit is achieved 
to within 0.1% by 20°K, and similar "saturation" ob- 

*Work performed under the auspices of the U. S. Atomic 
Energy Commission. 

t Present address, University of Washington, Seattle, Wash- 
ington. 

1 R. V. Pound and G. A. Rebka, Jr., Phys. Rev. Letters 3, 554 
(1959). 

*J. P. Schiffer and W. Marshall, Phys. Rev. Letters 3, 556 
(1959). 

*G. de Pasquali, H. Frauenfelder, S. Margulies, and R. N. 
Peacock, Phys. Rev. Letters 4, 71 (1960). 

* S. S. Hanna, J. Heberle, C. Littlejohn, G. J. Perlow, R. S. 
Preston, and and D. H. Vincent, Phys. Rev. Letters 4, 177 (1960). 



tains for the fraction /' of recoil-free absorption by an 
absorber of Fe 67 in Fe metal. Further cooling will 
result in a negligible increase of / and /'; cryogenic 
studies of the system might appear unprofitable. 

At sufficiently low temperatures, however, a re- 
distribution of the populations of the Zeeman sublevels 
takes place, and the nuclei become polarized. 5 This 
polarization can be quantitively studied through the 
effect upon the hyperfine Mossbauer spectrum. While 
nuclear polarization in ferromagnets has been observed 
before, 6 the present technique offers certain advantages. 
In this type of experiment the magnitude and sign of 
the magnetic field at the Co nucleus may be determined. 
The effects of the nuclear polarization are the concern 
of this paper. 

II. Fe" IN Fe METAL 

The ground state of Co 57 , with a half -life of 270 days, 
decays by -fiT-electron capture and neutrino emission to 
the second excited state of Fe 57 . The decay is probably 
an allowed transition, since its value of log// =6 lies 

6 J. G. Dash, R. D. Taylor, P. P. Craig, D. E. Nagle, D. R. F. 
Cochran, and W. E. Keller, Phys. Rev. Letters 5, 152 (1960). 

6 M. J. Steenland and H. A. Tolhoek, Progress in Low-Tem- 
perature Physics, edited by C. J. Gorter (Interscience Publishers, 
Inc., New York, 1957), Vol. II, Chap. X, p. 292. 



326 



1117 



POLARIZATION OF Co 67 IN Fe METAL 



within the limits of log// for known allowed transi- 
tions, 7 and this agrees with accepted spin assignments 
and considerations based on the shell model. The 
nuclear spin 7 3 of Co 57 is 7/2, and its magnetic moment 
H 3 is 4.65 nm. 8 The second excited state of Fe 57 has 
spin 1 2 of 5/2 and a mean lifetime of about 9X 10 -9 sec. 9 
This state decays, by emission of a 123-kev gamma ray, 
to the first excited state of Fe 57 . A minor fraction (9%) 
of the decays involve a transition from the second ex- 
cited state directly to the ground state, with the 
emission of a 137-kev gamma ray. 10 The multipolarity 
of the 123-kev radiation is 96% Ml, 4% £2. 911 The 
first excited state has spin Ii = 3/2, and moment ^i of 
0.153 nm. 4 This state has a half-life of 1.1X10- 7 sec, 
and decays by pure Ml emission of a 14.4-kev gamma 
ray, to the ground state of Fe 57 . The spin of the ground 
state, Jo, is 1/2, and it has a moment mo of 0.0903 nm. 12 
Experiments on the Mossbauer effect have been used 
to obtain a detailed description of the first excited and 
ground states of Fe 57 in Fe metal. 4 Results of this study, 
together with the properties of the higher states, are 
shown in Fig. 1. We also show the normal order of sub- 
levels of Co 57 , deduced from preliminary results 5 of the 
study reported here. 

The hyperfine magnetic field H acting on equivalent 
nuclei in a ferromagnetic crystal has a single direction 
in space over the region of a ferromagnetic domain. 
Coupling between the nuclear magnetic moments and 
the hyperfine field removes the degeneracy of nuclear 
spin orientation and produces a set of equally spaced 
spin sublevels of energies &.Eus=ingn n H, —I<m<I, 
where m is the magnetic quantum number, g is the 
nuclear gyromagnetic ratio, and n„ is the nuclear 
magneton. In cubic Fe metal, all nuclei occupy equiva- 
lent lattice positions ; hence, there 'is a single preferred 
direction in space and a single set of energy sublevels 
for the Fe nuclei in each ferromagnetic domain. An 
unmagnetized sample has its domains oriented in 
several directions, such that there is no net spatial 
polarization of the entire sample, but all of the nuclei 
(excepting those in the neighborhood of imperfections 
and impurities, and possibly those near domain walls) 
have the same hyperfine level splittings. The splittings 
of the first excited and ground states of Fe 57 are greater 
than the linewidth of the 14.4-kev resonance radiation. 
Gamma-ray transitions between the 14.4-kev sublevels 
of magnetic quantum number m } - to the ground-state 



Co= 



7 B. L. Robinson and R. W. Fink, Revs. Modern Phys. 32, 117 
(1960). 

8 J. M. Baker, B. Bleaney, P. M. Llewellyn, and P. F. D. Shaw, 
Proc. Phys. Soc. (London) A69, 353 (1956). 

9 G. F. Pieper and N. P. Heydenburg, Phys. Rev. 107, 1300 
(1957). 

10 D. E. Alburger and M. A. Grace, Proc. Phys. Soc. (London) 
A67, 280 (1954). 

11 G. R. Bishop, M. A. Grace, C. E. Johnson, A. C. Knipper, 
H. R. Lemmer, J. Perez y Torba, and R. G. Scurlock, Phil. Mag. 
46, 951 (1955). 

a G. W. Ludwig and H. H. Woodbury, Phys. Rev. 117, 1286 
(1960). 



:28 ■*.■*'* 

- 3/2 fi =+4.65 nm 
--I/2 3 

--3/2 



,- 9 -»j-°; 



EC (ALLOWED) 



-1/2 
+1/2 
+3/2 



Ml 

...1. 






-,| 




-AE ] 



I, = 3/2 
/i,»-O.I53nm 
AE=l.07xlO" 7 ev 



V l/2 

fj. =+0.0903 n m 
AE„* 1.90 x I0" 7 ev 



»57 



Fig. 1. Energy level diagram of Co 67 and daughter nuclei. 

sublevels m k (m k = mj,nij±l), therefore result in a 
gamma-ray spectrum of six hyperfine components. The 
relative intensity of the transition (ntj —> m k ) is pro- 
portional to the probability Wjk specified by the rules 
governing magnetic dipole radiation. Figure 2 is a 
schematic diagram of the radiation, similar to a diagram 
given by Hanna et al} The radiation widths are sup- 
pressed, and relative intensities are appropriate to the 
case of unpolarized radiation from a source having no 
net magnetization. 12a Positions of the line centers are 
given in terms of Doppler velocity shifts (positive ve- 
locity taken as increasing separation between source 
and detector) equivalent to shifts from the energy 
difference between degenerate excited and ground 
states. The m values of the upper and lower state sub- 
levels are shown at the top and bottom of each line. 



>- 
1- 






i "« 


- -3/2 


3/2 


- 


1 

§" 1/6 










- 


-1/2 


1/2 


- 


z 
o 

H 

to 1/12 

z 
< 
<r 






1/2 -1/2 




























-1/2 


-1/2 


4/2, 


1/2 




1/2 


1/2 




w 


\z -i.( 


9 -Q« 


6 0.« 


)6 


3.( 


te 5.: 


2 



DOPPLER VELOCITY v mm /SEC 

Fig. 2. Schematic diagram of the recoil-free 14.4-kev radiation 
from Fe 67 in Fe metal. Individual linewidths are not shown. 
Relative transition probabilities are appropriate to the case of 
domains oriented at random. Energy displacements are in terms of 
Doppler velocity, positive velocity being taken as increasing sepa- 
ration between source and observer. Magnetic sublevel quantum 
numbers for the first exxited and ground states are shown at top 
and bottom, respectively, of each line. 






See "note added in proof.' 



327 



DASH, TAYLOR, NAGLE, CRAIG, AND VISSCHER 



1118 



An absorber of Fe 67 in Fe metal has a similar hyperfine 
pattern of resonant cross sections. If Fig. 2 is translated 
over an identical pattern, the overlaps at velocity dif- 
ferences v= v (source) — v (absorber) represent the ab- 
sorption dips obtained when a source is moved relative 
to an absorber, resulting in a Mossbauer-type intensity 
pattern. As a result of chemical or temperature differ- 
ences between source and absorber, the emission and 
absorption spectra are shifted relative to each other 
by a small Doppler velocity 8v. 13M This shift is not 
essential to the present study, and the relative velocities 
v will be understood to represent the displacements 
from 8v. 

m. THEORY 

Polarization and Depolarization 

The intensity of the emission line (mj — > m k ) is pro- 
portional to the transition probability Wjk and to the 
population pj of the sublevel at which the transition 
originates. We define Wj k as the normalized relative 
intensity 

Wj k = pfOjk/H pju>jk. ( 1 ) 

In thermal equilibrium the populations Pj are propor- 
tional to the Boltzmann factors of the nuclear sublevels. 
The nuclear spins of the first excited state of Fe", 
however, are not in thermal equilibrium. Gossard and 
Portis 16 have measured a spin relaxation time of 10 -4 
sec for Co 69 nuclei in Co metal, and one may expect the 
relaxation time to increase as T~ l at lower tempera- 
tures. 16 The relaxation mechanisms for Co and Fe 
nuclei in Fe metal should be quite similar to those in 
the Co metal, and we may therefore treat the spin 
populations of the 14.4-kev state of Fe" as unchanged 
during the 10 -7 sec state lifetime. Since the lifetime of 
the second excited state is even shorter than 10~ 7 sec, 
the pj of a source of 14.4-kev radiation are functions of 
the populations of sublevels in the 270-day Co 57 parent. 
The equilibrium population pi of the Co 57 sublevel, in 
the case of pure magnetic hfs, is given by the Boltz- 
mann factor : 



Pi=C expimtfT- 1 ), where £=gnnHkr 



(2) 



mi is the magnetic quantum member of the sublevel, 
T is the temperature, g/z» is the moment of the Co 57 
nucleus, H is the hyperfine magnetic field at the 
nucleus, k is Boltzmann's constant, and C is a constant. 
A Co 67 nucleus in the mi sublevel decays to the mj sub- 
level of the 14.4-kev state of Fe" with a probability 
Qij. The matrix Q is the product of Clebsch-Gordan 

u R. V. Pound and G. A. Rebka, Jr., Phys. Rev. Letters 4, 274 
(1960). 

14 B. D. Josephson, Phys. Rev. Letters 4, 341 (I960). 

16 A. C. Gossard and A. M. Portis, Phys. Rev. Letters 3, 164 
(1959). 

16 G. E. Pake, in Solid State Physics, edited by F. Seitz and 
D. Turnbull (Academic Press, Inc., New York, 1956), Vol. 2, 



matrices for the two transitions preceding the arrival at 
the 14.4-kev state, and is presented in the Appendix. 
We can therefore obtain the pj of the 14.4-kev state by 
summing contributions from the parent sublevels : 



p j= const Y, i piQu- 



(3) 



Substituting Eqs. (2) and (3) in Eq. (1), we obtain the 
relative intensity of an emission line (m ; —* m k ) of a 
source, 

W jk =w jk £ e m '«'W(£ ™>« L e m «' T Qij). (4) 

l jk I 

We have assumed that spin lattice relaxation causes 
negligible depolarization of the spins during the K 
capture and subsequent gamma emissions. A second 
possible mechanism for depolarization is due to per- 
turbations by extranuclear fields resulting from K 
capture. A study of the angular distribution of the 123- 
kev radiation from partially aligned Co 67 nuclei in a 
Tutton salt 11 indicated considerable depolarization of 
the second excited state, presumably as a result of the 
AT-capture process. Depolarization to the extent ob- 
served in the Tutton salt would cause a marked de- 
crease in the population asymmetries. We believe at the 
outset, however, that the large electron mobilities in 
the Fe metal provide a rapid extinction of the per- 
turbing fields, making depolarization much smaller 
than in the salt. 

Transmitted Intensities 

We consider a resonance emission spectrum com- 
posed of several lines of Lorentzian shape, each line 
having the width T. The relative intensity of a line 
centered at energy E jk is W jk . When the source of 
radiation is moving away from the observer at speed v, 
the intensity distribution of the (mj—^mk) line is 
given by 

/r/27r 

*>*(») = . (5) 

(E-E } k+Ej k v/c¥+r>/4 



where / is the total intensity, 

I=T.Wjkf $j k (v)dE. 
ik J 



(6) 



The hyperfine emission spectrum of a source can be 
analyzed by filtering the radiation through an absorber 
containing ground-state nuclei: in the present case, 
Fe 67 . Absorbing nuclei are excited from ground-state 
sublevels to sublevels of the first excited state, and the 
transitions {m k > — > trig) have relative "intensities" W k 'y, 
where the primed symbols represent the absorber. The 
hyperfine resonant absorption cross section is composed 
of lines of cross section W k 'j>(Xk'j', where 



rr 2 /4 



(£-£* T ) 2 +r 2 /4 



(7) 



328 



1119 



POLARIZATION OF Co 67 IN Fe METAL 






and a is the total resonant absorption cross section. 
If such an absorber is placed between a source of radia- 
tion as is represented by Eq. (5), the intensity trans- 
mitted at relative Doppler speed v is 



I t (v) = ZW, 






Xexpl-naf £ W vr *vr¥Et (8) 

k'y 



where n is the total number of atoms/ cm 2 in the ab- 
sorber, a is the abundance of the isotope which absorbs 
resonantly, and /' is the fraction of recoil-free resonant 
absorptions. 17 

The overlap integral of Eq. (8) is implicitly limited 
to the case of unpolarized spectra obtained with un- 
magnetized sources and absorbers. In the event of a 
net magnetization it is necessary to take account of the 
relative orientations of the magnetic fields acting on 
the source and absorber nuclei. We will, however, limit 
this treatment to the case of unpolarized radiation, 
such as is represented in Fig. 2. The integral in Eq. (8) 
can be solved in closed form for two special cases: 
either perfect overlap of emission and absorption lines, 
or for no overlap. The latter case is equivalent to no 
resonant absorption, the transmitted intensity then 
being given by Eq. (6). In the case of perfect overlap, 
when an emission line energy E jk is Doppler shifted so 
that Ejk(l — v/c) = Ek'j>, the transmitted jk line in- 
tensity It(jk) can be written 



Itijk)- 



where y=2(E- 
Eq. (9) is 



I r <° dy r-W k .?x-l 

-W jk \ exp 

7T J-ool+y 2 L 1+y 2 J 



(9) 



I t (jk)=IW jk J (iW k ^x/2) exp(-W k . r x/2), (10) 

where J is the Bessel function of zeroth order. We shall 
assume that the 14.4-kev spectrum given in Fig. 2 
represents both the emission spectrum of the source 
and the absorption pattern of the absorber. The over- 
lap, or Mossbauer, pattern of such a source-absorber 
combination has several discrete Doppler speeds V at 
which the transmitted intensity due to all emission 
lines can be expressed in terms of the two special cases 
given above. Although the Lorentzian form of the lines 
vanishes only at infinity, a separation of 5r between 
source and absorption line energies is sufficient to 
reduce the resonant absorption to less than 1% of that 
at perfect overlap. We shall adopt this separation as 
a practical criterion for the absence of overlap. There 
are then four speeds V at which we can evaluate the 
transmitted intensity with good accuracy : These speeds 
are 2.23, 6.18, 8.41, and 10.46 mm/sec. The overlap and 

17 P. P. Craig, J. G. Dash, A. D. McGuire, D. Nagle, and R. D. 
Reiswig, Phys. Rev. Letters 3, 221 (1959). 



no-overlap contributions can be distinguished by a func 
tion Av having the properties : 

A v = 1, when E jk {\ - V/c) = E k >y, 



Av=0, when \E jk (l-V/c)-E k > j >\>5T. 



(ID 



The intensity transmitted at one of these discrete 
speeds can then be obtained by summing the con- 
tributions of overlapping and nonoverlapping lines : 

I t (V) = I £ W jk 

jkk'i' 

X{l-AK[l-/o(i^ fc 'W2)exp(-^ r x/2)]>. 

(12) 

The hyperfine radiation is, in practice, associated with 
a broad background of nonresonant gamma rays, and 
the absorber has a certain amount of nonresonant ab- 
sorption. The nonresonant background to the Moss- 
bauer pattern can be formally eliminated by comparing 
transmitted intensities at speed V with the intensity 
transmitted at high speed. The results of our experi- 
ment are expressed as a ratio, 

R(v)=u~-i<(vn/u~-i<(-vn (i3) 

where /«, is the intensity transmitted at speeds high 
enough so that no lines overlap, and h(V) and I t (— V) 
are the intensities transmitted at +V and —V, re- 
spectively. The explicit dependence of R(V) on the 
relative intensities of source and absorber lines is 
obtained by substituting Eq. (12) in Eq. (13) : 

R(V)= Z A v W ik K k . r {x)/ Z A-vW jk K k .j.(x), (14) 

jkk'i' jkk'j' 



E k .y)/Y, and x=naf'a. The solution of where 



tf*v(*) = l-J (iW k ^x/2) exp(-W k >i>x/2). 

We wish to obtain the explicit dependence of R(V) on 
the temperature of the source or of the absorber. Each 
case will be treated separately in the following sections. 

Cold Source 

We assume the absorber to be sufficiently warm so 
that we can neglect differences between sublevel popu- 
lations in the absorber: Wvy = Wk'j'/2~lk'j'Wk'i'. The 
temperature dependence of R(V) then arises from the 
differences in Boltzmann factors of the Co 67 sublevels 
and their influence on relative line intensities, Eq. (4). 
Before making the indicated substitution for W }k in 
Eq. (14), we note a symmetry property of the hyperfine 
spectrum, Fig. 2 : If emission line (j,k) overlaps absorp- 
tion line (k',f) at relative velocity V, then (— j, —k) 
overlaps (— k', — /) at velocity —V. Furthermore, the 
transition probabilities of symmetric lines are equal : 

Wjk = w-j- k , and w k '}' = W- k --y. 

Making use of these relations, substituting Eq. (4) in 



329 



DASH, TAYLOR, N A G L E , - C R A I G , AND VISSCHER 



1120 



-5 5 
>" 



O 4 





10.64 mm/ seo 


■ '1 1 


_ 


9.40 






3.60 


8.41 mm/see 


5.90 


" 






_ 






_ 




6. 1 8 mm /sec 


■ 2.79 




1.80 


««*' 2^23 mm/ sec 




- 






" 




1 1 


" 



20 30 40 50 

ABSORBER THICKNESS IN UNITS OF x 

Fig. 3. Calculated temperature coefficients of several relative 
absorption-emission velocities as a function of the thickness pa- 
rameter x. 



Eq. (14) yields 



R(V) = 



E A v K k >Ax)w jk j: e m «> T Qij 

jkk'j' i 

E AvK k 'Ax)u> jk Ze m ^ T Qi-j 
jkk'j' i 

£ v AvK k >Ax)w jk £ e m «' T Qii 

E LyK vr {x)w ik E e—f'Qti' 

jkk'j' i 



(15) 



since @y=()_i_y. Equation (15) takes a simple form at 
relatively high temperatures, when I-/T<K1. Expanding 
the Boltzmann factors to first order in %/T, we obtain 
the high-temperature approximation, 



where 



R(V)=l+a(V,x)!;T-\ 



(16) 



2 E Ar£*'j' (*)»,* E fnjQi/ 

a(P» = . (17) 

E &vKk>i'(x)wjkT,Qij 
jkk'j' i 

The decay from the first excited state to the ground 
state of Fe 67 is pure Ml. There is, however, a mixture 
of M 1 and E2 gamma rays in the decay from the second 
to first excited states. If we assume that the mixture 
is incoherent, as for the unpolarized spectra from un- 
magnetized sources, the temperature coefficient will 
have a similar mixture : 

R(V)=l+[p.96a 1 n(V,t)+0.04a E *(.Vj)lST-\ (18) 

where ajif i and a^2 are the coefficients corresponding to 
Eq. (17) with the proper matrix elements Qv for Ml 
and E2 radiation, respectively. 



The temperature coefficient a M \{V,x) for each speed 
V is shown as a function of the thickness parameter in 
Fig. 3. Coefficients a E i(V,x) are approximately one- 
half to one-third of the corresponding factor for the 
Ml radiation: The coefficient representing all of the 
transitions can therefore be estimated as 98% of the 
a m (V,x). 

Cold Absorber 

The temperature dependence of R(V) for a warm 
source and a cold absorber is related to the splitting 
of the ground-state levels of Fe 67 in the absorber, and 
is independent of the hfs of the ground state of Co 67 . 
Therefore, this case does not depend on the matrix 
elements Qi,, and the analysis is accordingly simpler. 
Also, no depolarization or coherence effects are present 
as considered above. Since we have not investigated 
this arrangement experimentally, we will only approxi- 
mate the temperature dependence. The approximation 
considered is that of a thin absorber, %Wwj'/2<£i. Ex- 
panding K k >j>{x) to first order in x, and expressing the 
relative strength W k >j> of an absorption line in terms 
of the population p k > of the ground-state sublevel, 



K v A%y=*xW vt ./2-- 






The most convenient overlap speeds for exploring the 
dependence of R(V) on absorber temperature are those 
at 7=6.18, 8.41, and 10.46 mm/sec. At these moder- 
ately strong absorption dips, all overlaps at + V are due 
to absorption lines arising from the + 1/2 ground-state 
sublevel, and all at — V originate from the — 1/2 sub- 
level. Equation (14) reduces, for these cases, to the 
particularly simple form : 



R(V) = p i /p. i =ex I >aoT-^, 



(20) 



where £ & is the energy splitting of the ground-state 
sublevels. 

TO H e TRANSFER Ht s MANOMETERS 
GAS AND VACUUM / AND PUMPS 



KEL-F SPRINGS 





SOURCE 
-SUPPORT TUBE 
(COPPER) 



MOTION 
.001' ALUMINIZED MYLAR 
Fig. 4. Schematic diagram of the experimental arrangement. 



330 



1121 



POLARIZATION OF Co" IN Fe METAL 



IV. EXPERIMENTAL APPARATUS 

A schematic diagram of the experimental arrange- 
ment is shown in Fig. 4. The cryostat has a somewhat 
unconventional design in that no liquid extends down 
to the height of the sample. The source is thermally 
protected by three concentric copper shields, the tops 
of which are maintained at liquid N 2 , liquid He 4 , and 
liquid He 3 temperatures. These features permit the 
14-kev radiation to leave the cryostat with an attenua- 
tion due only to thin windows in the shields; namely, 
0.001-in. Al at He 3 temperatures, 0.001-in. aluminized 
Mylar plastic at 1.4°K and 80°K, and 0.020-in. Be at 
room temperature. The source was connected directly 
to the He 3 reservoir via a heavy copper tube held 
rigidly in place within the He 4 shield by means of com- 
pressed leaf springs made of Kel-F plastic. The flange 
at the top of the He 4 shield was sealed to the He 4 
reservoir by means of a Sn-In O-ring. 

The desired temperature was reached and main- 
tained by controlling the pumping rate on the He 3 
bath. Temperatures were determined from the observed 
He 3 bath vapor pressures and a carbon resistance 
thermometer attached to the source holder. 

The 10-millicurie source was prepared in the manner 
described by Pound and Rebka. 1 Co 67 was plated from 
a Co 57 Cl 2 solution on a 0.007-in. thick of Armco iron 
which was then annealed in vacuum for approximately 
one hour at 900°C to cause diffusion of the Co 67 into 
the Fe lattice. The thin sheet was soft soldered to a 
copper holder for rigidity and good thermal contact. 

Absorbing foils of Fe 67 metal were prepared either by 
plating the enriched isotope on 0.1-mil Ni foil or by 
rolling sheets of the enriched Fe metal. The rolled foils 
were made by a technique used by Karasek of the 
Argonne National Laboratory. 18 A button of 78% Fe 67 , 
22% Fe 66 , was rolled to approximately 1-mil thickness 
in successive stages during which the sheets were 
annealed several times. The 1-mil sheet was further 
reduced by "pack rolling" between ferrotype plates, to 
approximately 0.1 mil. The foil was clamped between 
thin sheets of Mylar plastic by a soft iron frame holder. 

The detector consisted of a 1-mm thick Nal(Tl) 
crystal sealed to an RCA 6342 photomultiplier tube. 
Scintillation pulses corresponding to energies in the 
neighborhood of 14 kev were counted by means of a 
Franklin Model 358 amplifier and single-channel ana- 
lyzer, a modified Berkeley Model 7161-3 counter and a 
digital recorder. 

The absorber foils were mounted on a sliding carriage 
whose mean position was 20 cm in front of the scintil- 
lator crystal. A cable system drove the carriage at fixed 
linear speeds over a 6-cm horizontal path. This drive 
consisted of a 1/25-hp synchronous motor, worm gear 
speed reducers, and a 40-speed lathe gear box. Micro- 
switches at each end of the carriage path reversed the 
motor, and after an initial absorber travel of about 1 



S. S. Hanna (private communication). 



cm, the microswitches automatically reset and started 
the counter. The timing interval on the counter could 
be preset to four significant figures, so that counting 
could be made over almost the same absorber path at 
any given speed. This procedure averaged the measured 
intensities over nearly the same path length for positive 
and negative velocities, and was found to be necessary 
because of a 1% difference in counting rates between 
the extremes of absorber travel. The gear box and three 
interchangeable worm gear reducers allowed a choice of 
120 Doppler speeds, ranging from 0.0143 to 14.94 
mm/sec at a motor speed of 1800 rpm. Intervals be- 
tween the speed settings were adequate to explore the 
shape of the central absorption peak, but were in some 
cases too coarse to examine the details of other lines. 
We therefore used a vernier control on the speed by 
running the synchronous motor at power frequencies 
between 50 and 70 cps. The variable frequency power 
was generated by two audio oscillators and a power 
amplifier; the two oscillators were used alternately for 
positive and negative absorber velocities. This scheme 
made compensation possible for the small temperature 
and chemical shifts of the resonance patterns by setting 
the oscillators to different frequencies. The oscillator 
frequencies were measured and found to be stable to 
better than 0.1%. A 100-kc/sec crystal driven chrono- 
graph and optical gating system showed the absorber 
speeds to be uniform and stable to better than 0.03% 
over the whole length of travel when the motor was 
driven by the 60-cps power line. When two frequencies 
were used to drive the motor, the path length for the 
velocity corresponding to the lower drive frequency was 
slightly shorter because the preset timer on the counter 
was not alternated : The effect on the data is discussed 
in the next section. Unwanted relative motion of the 
source with respect to the absorber due to lateral 
vibration of the source within the cryostat was reduced 
by means of the Kel-F spacers. With this internal 
bracing, the assembly approached the rigidity of the 
outer casing of the cryostat; this in turn was fixed to 
the platform which supported the absorber carriage. 
Rigidity and good thermal isolation of the colder re- 
gions of the cryostat are somewhat incompatible; it 
was necessary to reach a compromise between the two 
extremes. 

Experimental Results 

The absorbers used were a plated foil of 2 mg/cm 2 Fe 57 
and a rolled foil of 1.73 mg/cm 2 Fe 67 . Although both 
foils had comparable resonance absorptions, the rolled 
foil was better in two respects; it had a narrower line 
and a smaller resonance pattern shift when both source 
and absorber were at room temperature. This shift was 
approximately 1.2X10 -2 mm/ sec, equivalent to a frac- 
tional resonant frequency difference between source and 
absorber of Av/v=4XlO~ l& , and could be accounted 
for by a difference between the characteristic Debye 



331 



DASH, TAYLOR, NAGLE, .CRAIG, AND VISSCHER 



1122 




Fig. 5. Experimental overlap pattern of Fe 67 in Fe metal, for 
Doppler speeds between and 3 mm/sec. The source is Co 67 in 
Fe metal at 4.5°K, and the resonant absorber is a 1.7-mg/cm 2 Fe 67 
rolled foil at room temperature. Intensities correspond to total 
counting rates of gamma rays having energies between 10 and 
18 kev. 



temperatures of source and absorber of 6°K. 19 The 
central absorption peak of the rolled foil had a width 
at half height of 0.42 mm/sec and a depth 53% (un- 
corrected for background) below the intensity at high 
velocities. This width is approximately two times that 
expected for a thin absorber, and corresponds closely 
to the width expected for the thick foil used. The first 
strong satellite absorption lines at 2.23 mm/sec were 
well resolved, and had the theoretical ratio, 0.57, of 
depth compared to the central peak. Upon cooling the 
source to low temperatures, the central resonance peak 
broadened to 0.60 mm/sec. The shift is in close agree- 
ment with that observed previously for an absorber at 
room temperature and a source at the temperature 
of liquid air. 13 The increased resonance width can be 
attributed to differences between the hfs of the source 
and absorber, due to the variation of hfs with tempera- 
ture; this effect has been studied in more detail at 
higher temperatures, 2021 and will not be discussed here. 
The resonance pattern obtained with the rolled foil 
and with the source at 4.5 °K is shown in Fig. 5. Vibra- 
tion associated with the accelerations at the ends of the 
absorber travel became excessive at higher speeds; 
consequently, the first strong satellite lines were judged 
most suitable for examining the intensity asymmetry 
at low temperatures. 

Preliminary experiments 6 conducted with the present 
source required the application of a large external mag- 
netic field in the plane of the source foil. An appreciable 
remanent magnetization of the source could possibly 
lead to errors in the present work if, in addition, the 

19 R. V. Pound and G. A. Rebka, Phys. Rev. Letters 4, 335 
(1960). 

20 S. S. Hanna, Proceedings of the Allerton Park Conference on 
the Mossbauer Effect, University of Illinois, 1960 (unpublished), 
pp. 39-40; D. H. Vincent, R. S. Preston, J. Heberle, and S. S. 
Hanna, Bull. Am. Phys. Soc. 5, 428 (1960). 

21 D. E. Nagle, H. Frauenfelder, R. D. Taylor, D. R. F. Cochran, 
and B. T. Matthias, Phys. Rev. Letters 5, 364 (1960). 



absorber were magnetized. The latter condition could 
result from the method of preparation of the absorber. 
A combination of the two circumstances would change 
the relative intensities of the absorption dips at the 
several source-absorber speeds V, as a result of the net 
polarization of individual spectral components. 4 To a 
first-order approximation, a net polarization does not 
influence the relative contributions of individual lines, 
and hence, the temperature dependence of R(V). 
Nevertheless, an experimental check of the net polariza- 
tion of the spectra was made; the absorber orientation 
was rotated by 90 degrees about an axis normal to its 
plane. This rotation caused no perceptible changes in 
transmitted intensities, ensuring that the spectrum of 
Fig. 2 was appropriate to the experiment. 

Experimental values of R (2.23 mm/ sec) for tempera- 
tures between 4.5° and 0.85°K are shown in Fig. 6. 
Data points were taken over a period of several hours 
for each temperature, in order to accumulate the neces- 
sary number of counts, ~10 7 , for adequate statistics. 
That the line does not pass through 1.00 at 1/T=0 is 
probably due to a geometry effect. The counting rate 
with the absorber placed at the extremes of the normal 
travel was shown to be slightly different ; as also noted 
earlier the absorber path length was slightly different 
for positive and negative velocities in this particular 
experiment. A systematic 0.3% change in the counting 
rate at one of the satellites used in obtaining the ratio 
would shift the ordinate in Fig. 6 by 0.015. 

The experimental slope, dft/dr-^ 0.03 13±0.0021 
obtained by a least squares analysis is directly propor- 
tional to the magnitude of the field H at the Co 57 
nuclei. In order to deduce H, it is necessary to evaluate 
the coefficients in Eq. (18) for the actual foil thickness. 

The total resonance cross section is given by the 
formula 17 

2I{+1 

<t=2ttX 2 ( ) = 1.48X10- I8 cm 2 , (21) 

2/o+lVl+c 



■lVl+a/ 




Fig. 6. Intensity ratio R for the 2.23-mm/sec resonant absorption 
dip at several temperatures between 4.5°K and 0.85°K. 



332 



1123 



POLARIZATION OF Co 67 IN Fe METAL 






where X is (27r) _1 times the wavelength of the 14.4-kev 
radiation, I\ and I are the spins of the excited and 
ground states, respectively, and 22 a =15 is the internal 
conversion coefficient. The calculated thickness pa- 
rameter for the 1.73-mg/cm 2 foil is x= 27. The effective 
thickness for the absorption lines having the intensity 
factors w*'.,-'=l/12, 1/6, and 1/4 which overlap in the 
2.23-mm/sec resonance are therefore xwk' } '/2=1.13, 
2.26, and 3.39,. respectively. The temperature coeffi- 
cients are calculated to be 

a M1 (V=2.23 mm/sec, *= 27) = 2.19, 
a £2 (F=2.23 mm/sec, z=27)=1.13. 

The resulting formula for the ratio of intensities is, 
by Eq. (18), 

R(V)=l+2.15ZT-\ (22) 

Comparison of Eq. (22) and the experimental value of 
R(V) leads to the measured value of the Co 67 level 
splitting, 

£=(14.6± 1.0) X lO-^K. 

The hyperfine magnetic field H corresponding to this 
splitting is 300±20 kilogauss. 12 * 

V. DISCUSSION 

The hfs magnetic field H acting on Co nuclei at low 
concentrations in Fe metal has been measured previ- 
ously by other methods. Table I lists the values obtained 
to date. All measurements were made at low tempera- 
tures. The experimental uncertainties in all of the 
determinations are probably within a factor of 2 of the 
7% estimated for the present work. There is no evident 
disagreement among the several measurements. Since 
the earlier studies could not be subject to the depolari- 
zation mechanisms discussed earlier in this paper, it is 
apparent that depolarization does not play an important 
role in the present technique. It is clear that we have 
observed no major depolarization such as occurs for 
Co 67 in a Tutton salt. 11 We can conclude that the per- 
turbing fields which are considered to be responsible 

Table I. Hyperfine field H at Co nuclei in Fe metal. 

H 
Reference Method (kilogauss) 

Present work 300 

(23) Specific heat 320 

(24) Specific heat 315 

(25) Gamma-ray (Co 60 ) anisotropy 350 



22 H. R. Lemmer, O. J. A. Segaert, and M. A. Grace, Proc. Phys. 
Soc. (London) A68, 701 (1955). 

23 V. Arp, D. Edmonds, and R. Petersen, Phys. Rev. Letters 3, 
212 (1959). 

24 N. Kurti, Suppl. J. Appl. Phys. 30, 2155 (1960). 

26 A. V. Kogan, V. D. Kul'kov, L. P. Nikitin, N. M. Reinov, 
I. A. Sokolov, and M. F. Stel'makh, J. Exptl. Theoret. Phys. 
(U.S.S.R.) 39, 47 (1960) [translation : Soviet Phys.— JETP 12(39), 
34 (1961)]. 



for the Tutton salt results arise after the -fiT-capture 
decay of the Co 87 , and are probably due, in the Tutton 
salt, to long-lived holes in the outer electron shells. 
These holes are filled rapidly by the conduction elec- 
trons of the metal, in times that are short compared to 
the 10~ 8 -sec lifetime of the 136-kev state of Fe 57 . 

Finally, we note that the magnitude of H for Co in 
Fe metal is much closer to the field value of 333 kilo- 
gauss for Fe in Fe metal 4 than to the value 219 kilo- 
gauss for Co in Co metal. 15 It is not surprising that for 
these materials the effect of environment appears to 
dominate those interactions which may be ascribed to 
the individual atoms. Co differs from Fe in that it has 
one additional 3d electron, which is probably accepted 
into the unfilled 3d band of the Fe metal, thus leaving 
the Co nucleus in an environment characteristic of the 
surrounding Fe. 

ACKNOWLEDGMENTS 

We gratefully acknowledge the contributions of 
several people. L. Wilets of the University of Washing- 
ton helped us to clarify our understanding of details 
of the decay scheme. R. Keil provided the thin rolled 
absorbers, and J. M. Dickinson assisted in the prepara- 
tion of samples. W. E. Keller and D. R. F. Cochran, who 
collaborated on the initial experiments, encouraged and 
assisted us in the present work. R. R. Rylander con- 
structed portions of the apparatus, and R. Hanft 
assisted in many of the measurements. 

APPENDK 

Here we calculate the coefficients Qu which were 
introduced in Eq. (3). Qij is the probability that, if the 
Co 67 nucleus initially has magnetic quantum number 
mi, the decay will proceed to the Fe 67 first excited state 
with magnetic quantum number tn : : It depends, in 
addition to the spins of the nuclear states involved, 
upon the character of the K capture, and on the multi- 
polarity of the 7 ray emitted in the transition between 
the second and first excited states. 26 As has been dis- 
cussed in the text, the K capture is almost certainly 
Gamow-Teller allowed, and we will calculate the co- 
efficients for both of the possible multipolarities, 
namely, Ml and E2. In case the radiation is not pure, 
but, as is realized for this y ray, is a mixture of Ml and 
E2, there will in general be interference between the 
two components. However, the interference term van- 
ishes when averaged over angle, so that if the source is 
unmagnetized it will have no effect, and the result ob- 
tained by adding the two contributions [see Eq. (18)] 
is correct. 

If we denote the wave functions of Co 67 and the 
second and first excited states of Fe 57 by ^7/2"", ^5/2 m ', 
and ^3/2 my , respectively, then the cobalt decay may be 

26 It also depends on depolarizing forces, if any, which act in 
the intermediate state. We calculate it on the assumption that 
there are none. 



333 



DASH, TAYLOR, NAGLE, CRAIG, AND VISSCHER 1124 

Table II. Q h for £=1 (Ml y ray). Table III. Q tj for Z.= 2 (El y ray). 



\f», 










\wy 










Wj\ 


3/2 


1/2 


-1/2 


-3/2 


mi\ 
7/2 


3/2 
3/7 


1/2 
4/7 


-1/2 



-3/2 


7/2 


1 














5/2 


4/7 


3/7 








5/2 


24/49 


9/49 


16/49 





3/2 


2/7 


4/7 


1/7 





3/2 


22/49 


4/49 


19/49 


4/49 


1/2 


4/35 


18/35 


12/35 


1/35 


1/2 


12/35 


38/245 


72/245 


51/245 


-1/2 


1/35 


12/35 


18/35 


4/35 


-1/2 


51/245 


72/245 


38/245 


12/35 


-3/2 





1/7 


4/7 


2/7 


-3/2 


4/49 


19/49 


4/49 


22/49 


-5/2 








3/7 


4/7 


-5/2 





16/49 


9/49 


24/49 


-7/2 











1 


-7/2 








4/7 


3/7 



represented by 

fr/2 ml -» Em' CO/2, mi 1 5/2, m' ; 1, m t -m') 

x^ 6/ 2 m 'xr , - ro ', (Al) 

where C is the usual Clebsch-Gordan coefficient and x 
is a triplet S-wave function describing the emitted 
neutrino plus the absorbed electron. In turn, the decay 
of the second excited state is written 

^ 6 /2 m ' — » 2~L C(5/2,m'\3/2, m,; L. m'—m 3 ) 

Xh/2 m 'yL m '- m '\ (A2) 

where y L represents the emitted y ray of multipole order 
L. Upon substituting (A2) into (Al), 

fan-^Jl H C(7/2, mi\5/2m'; 1, nti—m') 

XC(5/2, w' 1 3/2, mj; L, m'-m,) 

Xxi m ~ m 'yL m '~ mi h/2 mi , 

we find that the sum of squares of the contributions to 
the coefficient of ^ 3 /2 m; is 



Qu=?: m >\C(7/2,mi\S/2,m';l,mi-m') 
XC(5/2,m'\3/2,mj 



L, m'- 



h)Y 



The Clebsch-Gordan coefficients may be easily calcu- 
lated, or found in tables, and the sum (which never 
contains more than 3 terms) evaluated numerically. 
The results are shown in Table II for L=\ (dipole y 
ray), in Table III for L=2 (quadrupole y ray). 

Note added in proof. Recent examination of the hyper- 
fine spectra of source and absorber by an unsplit ab- 
sorber and source, respectively, has yielded relative in- 
tensities in the ratio 3 : 3.2 : 1, indicating that the samples 
were partially magnetized. Calculations based upon the 
revised spectrum increases the deduced value of the 
hyperfine field at Co 57 nuclei to 375 kgauss. Further 
deviation of the actual intensity distribution from the 
distribution assumed in the text does not cause a further 
increase in the calculated field. Uncertainty in the inter- 
mediate magnetic history of the specimens prevents 
specifying the hyperfine field more precisely within the 
limits of 300 and 375 kgauss. Experiments now in 
progress should resolve the uncertainty in the near 
future. 



COMMENTS AND CORRECTIONS 



The authors of the papers listed below have requested that the 
following corrections to their papers be noted: 



Diffusion des photons sur les atomes et les noyaux dans les cristaux, 
by C. Tzara« J. phys. radium, 22, 303 (1961) 



>5 



205 2nd col., 4th line under head: For "[4] " read "[3] 
2nd col., 11th line under head: For "[5]" read "[4] 

206 1st col., 11th line: For "dN(w)/aw" read "dN(u})/du>" 
1st col., 13th line: For "(5)" read "[4]" 

207 1st col., line above eq. (1): For "[6]" read "[5]" 
Eq. (1): The numerator should read 

(Os}|e iKlU/K Hn s }){n s }le iK = u / K | { a s }> 

Last equation on page: 

Numerator should start 1( { etc. 

208 1st col., eq. (3): Square the whole term in the first line of 

the equation 
1st col.: Equation below (3) should read 

k is = K i € s( 2MNn ^s)" 1/2 



Polarized Spectra and Hyperfine Structure in Fe 57 , by S. S. Hanna 
et al.^Phys. Rev. Letters, 4, 177 (1960) 

226 Figure 2: Reverse the signs of all the magnetic quantum 
numbers 

335 



336 



;OMMENTS AN 



RECTIONS 



#2 * .Line 33. Sente ;e should read ' 'Thus, the -.ings between 
lines 1 and 2, 4 and 5, and 5 and 6 should be equal to the 
splitting of the excited state." 



Temperature -Dependent Shift of y Rays Emitted by a Solid, by B. D, 
Josephson • Phys. Rev. Letters, 4, 341 (1960) 

252 1st col.: The last equation should read 

6E =(6H> = (6(p 2 i /2m i )) 

= -3mi<pf/2ml> 

= - {bm i /m i )T i = (E/miC 2 )^ 



Recoilless Rayleigh Scattering in Solids, by C. Tzara and 
R. Barloutaud«Phys. Rev. Letters, 4, 405 (1960) 

279 1st col.: Equation (1) should read 

x 



^T = ex P "2iS 



3 E r 



1 + 



st^u 



The Mossbauer Effect in Tin from 120°K to the Melting Point, by 
A. J. F. Boyle et al. •Proc. Phys. Soc. (London), 77, 129 (1961) 

291 Eqs. (6), (7), and (8): A minus sign should precede the 

right-hand side of each equation 

292 Eq. (10) should read 






fleet a review, W»h a engr 

3 1HW2 0E20H ltH3 



/ r sy 



-77-? 



c,;& 






r i 



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